Ion trap quantum processor

Transcription

1 Ion trap quantum processor Laser pulses manipulate individual ions row of qubits in a linear Paul trap forms a quantum register Effective ion ion interaction induced by laser pulses that excite the ion`s motion A CCD camera reads out the ion`s quantum state

2 Experimental setup Fluorescence detection by CCD camera photomultiplier atomic beam Photo multiplier CCD camera oven Laser beams for: Vacuum pump photoionization cooling quantum state manipulation fluorescence excitation

3 Quantum jumps: spectroscopy with quantized fluorescence P D monitor metastable level weak transition absorption and emission cause fluorescence steps (digital quantum jump signal) S S Quantum jump technique Electron shelving technique Observation of quantum jumps: D Nagourney et al., PRL 56,2797 (1986), Sauter et al., PRL 57,1696 (1986), Bergquist et al., PRL 57,1699 (1986)

4 Electron shelving for quantum state detection P1/2 1. Initialization in a pure quantum state D5/2 τ =1s Quantum state Fluorescence manipulation detection 40 SS1/21/2 Ca+ 2. Quantum state manipulation on S1/2 D5/2 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 times counts per 2 ms

5 Electron shelving for quantum state detection P1/2 D5/2 Fluorescence detection S1/2 Two ions: Spatially resolved detection with CCD camera: 1. Initialization in a pure quantum state 2. Quantum state manipulation on S1/2 D5/2 transition 3. Quantum state measurement by fluorescence detection 5µm 50 experiments / s Repeat experiments times

6 Quantum harmonical oscillator Extension of the ground state: Size of the wave packet << wavelength of visible light Energy scale of interest: harmonic trap

7 Laser ion interactions harmonic trap... 2 level atom e joint energy levels e,1 e,0 e,2 g,2 g,1 g g,0 Approximations: Ion: Trap: Electronic structure of the ion approximated by two level system (laser is (near ) resonant and couples only two levels) Only a single harmonic oscillator taken into account

8 External degree of freedom: ion motion harmonic trap... 2 level atom joint energy levels

9 A closer look at the excitation spectrum (3 ions) (red / lower) motional sidebands carrier transition (blue / upper) motional sidebands Laser detuning at 729 nm (MHz)

10

11 Sideband absorption spectra % g r o u n d s ta t e p o p u la t io n a fte r D o p p le r c o o lin g 0.8 n z = 1.7 a fte r s id e b a n d c o o lin g D P P D D e tu n in g δ ω ( M H z ) red sideband D e tu n in g δ ω ( M H z ) blue sideband

12 Coherent excitation on the sideband Blue sideband pulses: coupled system D,n 1 D,n... D,n 1... S,n 1 S,n S,n 1 Entanglement between internal and motional state! D state population

13 Single qubit operations Arbitrary qubit rotations: Laser slightly detuned from carrier resonance or: (z rotations by off resonant laser beam creating ac Stark shifts) Concatenation of two pulses with rotation axis in equatorial plane Gate time : 1 10 µs D state population Coherence time : 2 3 ms limited by magnetic field fluctuations laser frequency fluctuations (laser linewidth δν<100 Hz)

14 Addressing the qubits Paul trap 0.6 Excitation coherent manipulation of qubits electro optic deflector dichroic beamsplitter Fluorescence detection CCD Deflector Voltage (V) 8 10 inter ion distance: ~ 4 µm addressing waist: ~ 2 µm < 0.1% intensity on neighbouring ions

15 Pulse sequence: Generation of Bell states

16 Generation of Bell states Pulse sequence: Ion 1: π/2, blue sideband

17 Generation of Bell states Pulse sequence: Ion 1: π/2, blue sideband Ion 2: π, carrier

18 Generation of Bell states Pulse sequence: Ion 1: π/2, blue sideband Ion 2: π, carrier Ion 2: π, blue sideband

19 Bell state analysis 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 DD DS SD SS

20 Obtaining a single qubits density matrix (a naïve persons point of view) A measurement yields the z component of the Bloch vector => Diagonal of the density matrix Rotation around the x or the y axis prior to the measurement yields the phase information of the qubit. => coherences of the density matrix

21 Bell state reconstruction F=0.91 SS SD DS DD SD SS DS DD SS SD DS DD SD SS DS DD

22 Phase gate CNOT Both, the phase gate as well the CNOT gate can be converted into each other with single qubit operations. Together with the three single qubit gates, we can implement any unitary operation!

23 Quantum gate proposals with trapped ions...allows the realization of a universal quantum computer! Some other gate proposals by: Cirac & Zoller Mølmer & Sørensen, Milburn Jonathan, Plenio & Knight Geometric phases Leibfried & Wineland

24 Cirac Zoller two ion phase gate 1 2 D, 0 S, 0 ion 1 motion ion 2 S,D 0 S,D SWAP D,1 S,1 0

25 Cirac Zoller two ion phase gate 1 2 Phase gate using the motion and the target bit. ion 1 motion ion 2 S,D 0 S,D SWAP 0 Z

26 Cirac Zoller two ion phase gate 1 2 D, 0 S, 0 ion 1 motion ion 2 S,D 0 S,D SWAP SWAP Z D,1 S,1 0

27 Cirac Zoller two ion phase gate 1 2 Phase gate using the motion and the target bit. ion 1 motion ion 2 S,D 0 S,D 0 Z

28 Cirac Zoller phase gate: the key step D,1 D,0 S,1 S,0 A 2π pulse is applied to only one of ion (crystal)s states => only one states acquires a phase factor of 1. An additional Zeeman level can be used as the auxilary state. => gate is sensitive to magnetic field fluctuations!

29 How do you do with just a two level system??

30 Phase gate Composite 2π rotation:

31 A phase gate with 4 pulses (2π rotation) R(, ) R1, 2 R1 2,0 R1, 2 R on S, 0 2,0 D,1

32 A single ion composite phase gate: Experiment state preparation S,0, then application of phase gate pulse sequence π / 2 π / 2 π φ=π /2 ϕ=0 φ=0 ϕ=0 φ=0 π φ=π /2 ϕ=π / D5/2 excitation Time (µs)

33 Population of S,1> D,2> remains unaffected R(, ) R1 2, 2 R1,0 R1 2, 2 R1,

34 Testing the phase of the phase gate 0,S> π Phase gate 2 1 π (5) % 0.9 D5/2 excitation Time (µs)

35 Phase gate with starting in D,1> π π Blue Phase gate 2 π 2 1 D5/2 excitation Time (µs)

36 Cirac Zoller two ion controlled NOT operation ion 1 motion ion 2 S,D 0 S,D pulse sequence: ion 1 ion 2 SWAP SWAP control qubit 0 target qubit

37 Cirac Zoller CNOT gate operation SS SS DS DD SD SD DD DS

38 Measured truth table of Cirac Zoller CNOT operation input output

39 Superposition as input to CNOT gate prepare gate output detect

40 Error budget for Cirac Zoller CNOT Error source Magnitude Population loss Laser frequency noise (Phase coherence) ~ 100 Hz (FWHM) ~ 10 %!!! <n>bus < 0.02 <n>spec = 6 2% 0.4 % Laser intensity noise 1 % peak to peak 0.1 % Addressing error (can be corrected for partially) 5 % in Rabi frequency (at neighbouring ion) 3% Off resonant excitations for tgate = 600 µs 4% Laser detuning error ~ 500 Hz (FWHM) ~2% Total November 2002 ~ 20 % Residual thermal excitation

41 Error budget for Cirac Zoller CNOT Error source Magnitude Population loss Laser frequency noise (Phase coherence) ~ 100 Hz (FWHM) ~ 10 %!!! <n>bus < 0.02 <n>spec = 6 2% 0.4 % Laser intensity noise 1 % peak to peak 0.1 % Addressing error (can be corrected for partially) 5 % in Rabi frequency (at neighbouring ion) 3% Off resonant excitations for tgate = 600 µs 4% Laser detuning error ~ 500 Hz (FWHM) ~2% Total November 2002 ~ 20 % Residual thermal excitation

42 Testing the phase of the phase gate 0,S> π Phase gate 2 1 π (5) % 0.9 D5/2 excitation Time (µs)

43 Ramsey experiment Time Quantum algorithms can be viewed as generalized Ramsey experiments! Laser detuning (khz)

44 Phase coherence Gaussian modell yields a coherence time of 0.9 ms Now (in 2005) 2 ms are more typical.

45 Test of the motional coherence 0,S 1,S 1,D 0,D 1,S 0,S π/2 1,D 0,D uv light 0,D 1,D π 1,S 0,S π π/2 0,D 0,S 1,D 1,S ±π/2 Excitation to D5/2 state Time (ms) Time (ms) After 60 ms still 50 % contrast!

46 Error budget for Cirac Zoller CNOT Error source Magnitude Population loss Laser frequency noise (Phase coherence) ~ 100 Hz (FWHM) ~ 10 %!!! <n>bus < 0.02 <n>spec = 6 2% 0.4 % Laser intensity noise 1 % peak to peak 0.1 % Addressing error (can be corrected for partially) 5 % in Rabi frequency (at neighbouring ion) 3% Off resonant excitations for tgate = 600 µs 4% Laser detuning error ~ 500 Hz (FWHM) ~2% Total November 2002 ~ 20 % Residual thermal excitation

47 Intensity noise => Laser intensity noise: 0.03

48 Error budget for Cirac Zoller CNOT Error source Magnitude Population loss Laser frequency noise (Phase coherence) ~ 100 Hz (FWHM) ~ 10 %!!! <n>bus < 0.02 <n>spec = 6 2% 0.4 % Laser intensity noise 1 % peak to peak 0.1 % Addressing error (can be corrected for partially) 5 % in Rabi frequency (at neighbouring ion) 3% Off resonant excitations for tgate = 600 µs 4% Laser detuning error ~ 500 Hz (FWHM) ~2% Total November 2002 ~ 20 % Residual thermal excitation

49 Qubit Rotations with two ions Pulse sequence: carrier pulse error = Ineighbour neighbour = target Itarget θ, φ=0 adressing error 5% 2 excitation 1 0 typical adressing errors are in the range of 3 5% 0 4π 8π 12π rotation angle θ 16π 20π

50 Error budget for Cirac Zoller CNOT Error source Magnitude Population loss Laser frequency noise (Phase coherence) ~ 100 Hz (FWHM) ~ 10 %!!! <n>bus < 0.02 <n>spec = 6 2% 0.4 % Laser intensity noise 1 % peak to peak 0.1 % Addressing error (can be corrected for partially) 5 % in Rabi frequency (at neighbouring ion) 3% Off resonant excitations for tgate = 600 µs 4% Laser detuning error ~ 500 Hz (FWHM) ~2% Total November 2002 ~ 20 % Residual thermal excitation

51 Off resonant carrier excitation D,0 S,0 Problem: D,1 S,1 S D e x c ita t io n p r o b a b ilit y ω z = 2π 1.8 MHz carrier = 2π 1.09 MHz AC Stark shifts off resonant (carrier) excitation (spectator modes) in te r a c tio n t im e ( µ s ) A. Steane, C. F. Roos, D. Stevens, A. Mundt, D. Leibfried, F. Schmidt Kaler, R. Blatt, Phys. Rev. A 62, (2000)

52 Computation method 0,D 1,D Solve master equation! Schrödinger equation for 1 ion: 0,S 0,D 0,S 1,S 1,D iη 0 C1 C1 0 Δ iη 0 0 C2 d C2 i = dt C 3 0 iη 0 ωt 1 η 2 0 C 3 C4 C4 iη η2 0 Δ ωt η 0.03: Lamb-Dicke-parameter Δ: Laser detuning / 1,S 0,S 0,D 1,S 1,D n: Number of phonons / ωt : Trap frequency Hilbert space: 0,1,2 S,D,Daux 1 S,D,Daux 2 (27 dimensions)

53 Error budget for Cirac Zoller CNOT Error source Magnitude Population loss Laser frequency noise (Phase coherence) ~ 100 Hz (FWHM) ~ 10 %!!! <n>bus < 0.02 <n>spec = 6 2% 0.4 % Laser intensity noise 1 % peak to peak 0.1 % Addressing error (can be corrected for partially) 5 % in Rabi frequency (at neighbouring ion) 3% Off resonant excitations for tgate = 600 µs 4% Laser detuning error ~ 500 Hz (FWHM) ~2% Total November 2002 ~ 20 % Residual thermal excitation

54 Summary of the complications of the Cirac Zoller approach the gate is slow (off resonant excitations) => sensitive to frequency fluctuations / deviations. the gate requires addressing => trap frequency needs to reduced Use other gate types which do not require addressing. the gate is sensitive to motional heating Not really a problem yet.

55 Scaling of this approach? Problems : Coupling strength between internal and motional states of a N ion string decreases as (momentum transfer from photon to ion string becomes more difficult) > Gate operation speed slows down More vibrational modes increase risk of spurious excitation of unwanted modes Distance between neighbouring ions decreases > addressing more difficult > Use flexible trap potentials to split long ion string into smaller segments and perform operations on these smaller strings

56 Scaling ion trap quantum computers Easy to have thousands of ions in a trap and to manipulate them individually but it is hard to control their interaction! The solutions: solution: Kielpinski, Monroe, Wineland Cirac, Zoller, Kimble, Mabuchi Tian, Blatt, Zoller

57 Idea #1: move the ions around D. Wineland, Boulder, USA Kieplinski et al, Nature 417, 709 (2002) accumulator control qubit target qubit

58 Segmented ion traps as scalable trap architecture (ideas pioneered by D. Wineland, NIST, and C. Monroe, Univ. Michigan) Segmented trap electrode allow to transport ions and to split ion strings State of the art: Transport of ions 1 mm within 50 µs no motional heating Splitting of two ion crystal tseparation 200 µs small heating n 1 Architecture for a large scale ion trap quantum computer, D. Kielpinski et al, Nature 417, 709 (2002) Transport of quantum states, M. Rowe et al, quant ph/

59 or use a head ion I. Cirac und P. Zoller, Nature 404, 579 (2000) quantum optics and nano technology: scalability m o tio n head ta rg e t p u s h in g la s e r

60 Idea #2: coupling via photons P3/2 y vit Ca D5/2 S1/2 Transfer quantum state of the ion to cavity photon: qubit interface π/2 Create superposition photon state (STIRAP) Detect cavity output and ion state Choose coupling g by STIRAP detuning qubit 1 photonic channel J. I. Cirac et al., PRL 78, 3221 (1997) A. Kuhn et al., PRL 89, (2002) Controlled ion cavity interaction qubit 2

61 Idea #3: coupling via image charges

62 Idea #3: coupling via image charges + +

63 Idea #3: coupling via image charges + + Coupling in the khz range requires small traps (~20 µm) Very small currents can be measured => non invasive probe for conductors (superconductors) Connect to solid state qubits

64 Quantum information processing: Which ion is best? Ions with optical transition to metastable level: 40Ca+,88Sr+,172Yb+ P1/2 D5/2 τ 1 s τ =1s Sideband cooling 40 + Doppler cooling Qubit levels: S1/2, D5/2 Qubit transition: Quadrupole transition S1/2 D5/2 Ca SS1/21/2 Ions with hyperfine structure: 9Be+, 43Ca+,111Cd+, F +1 P1/2 S1/2 F F F Qubit levels: Hyperfine levels of ground state S1/2 Qubit transition: Raman transition between hyperfine levels

65 Disadvantage of optical quantum bits Ions with optical transition to metastable level: 40Ca+,88Sr+,172Yb+ P1/2 e> Doppler detection cooling SS1/2 g> 1/2 optical qubit metastable D5/2 τ =1s optical Sideband transition cooling 40 + Ca stable qubit manipulation requires ultrastable laser Coherence time limited by laser linewidth Ions with hyperfine structure: 9Be+, 43Ca+,111Cd+, hyperfine qubit detection e> g> hyperfine ground states qubit manipulation with microwaves or lasers

66 Hyperfine qubit + microwaves Single qubit manipulation with microwaves: detection hyperfine ground states well established technique (NMR), long coherence times Problems: Addressing of single qubit in ion string difficult Coupling between internal and motional states: Coupling constant neglegible!

67 Hyperfine qubit + microwaves + magn. field gradient? (F. Mintert, C. Wunderlich, PRL 87, (2001)) Apply magnetic field gradient to obtain state dependent potential: Coupling constant: (x0: size of ground state) Example: requires strong field gradients

68 Hyperfine qubit + Raman transitions Coupling between hyperfine states by Raman transition detection hyperfine ground states If both Raman beams are derived from the same laser, the laser line width ΓL is not important as long as ΓL <<. f1 f2 (acousto opt. modulators) Raman beam generation

69 Raman transitions: Three level system Laser frequencies Rabi frequencies Detuning from upper state After adiabatic elimination of upper level: Coupling between ground states Rabi frequency of Raman process

70 Raman transitions: Further effects Laser frequencies Rabi frequencies Detuning from upper state 2. The energies of states 1>, 2> are light shifted because of the interaction with 3> by In case of unequal light shifts, the transition frequency between 1> and 2> depends on the laser intensity. (laser intensity noise > decoherence) 3. Population in state 3> : photon scattering rate choose large detuning

71 200 GHz Raman transitions in real ions Be+ (used in Wineland group, Boulder) 1.25 GHz 313 nm 9 Raman beams tuned to frequency within P1/2 fine structure << fine structure: Constructive interference between P1/2 and P3/2virtual levels strong Raman coupling >> fine structure: Destructive interference between P1/2 and P3/2virtual levels weak Raman coupling, but longer hyperfine coherence times (only Raman scattering destroys hyperfine coherence, R. Ozeri et al, PRL 95, (2005))

72 Raman transitions in real ions 111 Cd+ Monroe group Univ. Michigan simple level structure (Raman beam generation technically challenging) P.Lee et al, Opt. Lett. 28, 1582 (2003) 43 Ca+ Steane group Oxford many levels more convenient laser wave lengths A. Steane, Appl. Phys. B 64, 623 (1997) Innsbruck see also: D. Wineland et al. Phil. Trans. R. Soc. Lond. A 361, 1349 (2003)

73 Raman transitions: motional sidebands Same Hamiltonian as for two level system, with Raman Rabi frequency Raman detuning from hyperfine transition difference of Raman laser phases effective Lamb Dicke parameter

74 Raman transitions: motional sidebands effective Lamb Dicke parameter copropagating Raman beams for excitation of carrier transitions (no sideband transitions) counterpropagating Raman beams for efficient excitation of sideband transitions or beams from different directions

75 Decoherence issues hyperfine (Raman) qubit optical qubit relative laser path length fluctuations (stable setup, insensitive gate operations) Raman photons scattered from virtual level (choose ion with big fine structure) laser frequency noise (ultrastable laser) decay of metastable state (long lived state) magnetic field fluctuations (screen magnetic fields, use field independent transition) fluctuations of control parameters

76 Entangling interactions: controlled phase gate Use Raman beams that couple the motional states (but not internal states) Raman beams form (moving) standing wave: spatial light shifts E P3/2 F P1/2 x F F = 2 F n=3 n=3 State dependent optical dipole force n=0 n=0 S1/2

77 Stretch mode excitation no differential force differential force differential force no differential force

78 Controlled phase gate Time evolution operator: Choose α(t ) such that

79 Phase space picture Boulder group : Gate fidelity: Gate time: 97% 7 µs (ca. 25/νCOM) D. Leibfried et al., Nature 422, 414 (2003) Theory: Milburn, Fortschr. Phys. 48, 9(2000) Sørensen&Mølmer

80 Geometric phase gate 1) coherent displacement along closed path will shift phase of the quantum state, phase independent of details like speed of traversal, etc. 2) the sign and magnitude of coherent displacements can be made internal state dependent (see e.g. Science 272, 1131 (1996)) no ground state cooling no individual addressing robust against small deformations of the path relative phase of successive displacements irrelevant G. J. Milburn et al., Fortschr. Physik 48, 801 (2000). X. Wang et al., Phys. Rev. Lett. 86, 3907 (2001).

81 D. Wineland (NIST) : Alumina / gold trap

82 C. Monroe (Michigan) : GaAs GaAlAs trap

83 Ideal Cirac Zoller phase gate ( J. I. Cirac und P. Zoller, PRL (1995)) 0, SD aux 0, DS 0, SS qubits , DS 0, SD π 1, SS 0, SS red sideband π pulse on ion #1 0,SS 0,SD 0,DS 0,DD 1, SD aux 2π 1, SD 1, SS 0, DS 0, SS red sideband 2π pulse on ion #2 0,SS 0,SD i 1,SS i 1,SD 0,SS 0,SD i 1, SS i 1,SD 1, DS π 1, SS red sideband π pulse on ion #1 0, SS 0, SD 0, DS 0,DD