Quantum teleportation

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1 Quantum teleportation "Deterministic quantum teleportation with atoms", M. Riebe et al., Nature 429, 734 (2004). "Deterministic quantum teleportation of atomic qubits", M. D. Barrett et al., Nature 429, 737 (2004). Motivation 1

2 Teleportation idea (Bennett 1993) ALICE measurement in Bell basis recover input state 1 class. communication unknown input state 2 3 rotation BOB Bell state Full formal procedure 2

Very simple description (see Bouwmeester et al.) Particle 2 and 3 are prepared in an entangled state where one particle is always in the "opposite" state of the other.

3 Very simple description (see Bouwmeester et al.) Particle 2 and 3 are prepared in an entangled state where one particle is always in the "opposite" state of the other. When the result of the Bell measurement between 1 and 2 is then 2 is in the "opposite" state of the unknown state of 1. Thus particle 3, being in the opposite state to 2, will be in the same state as 1. If the result of the Bell measurement is a different state, the appropriate rotation has to be applied to particle 3. Teleportation of atomic qubit states Figure from Kimble, Nature N&V 3

4 Teleportation protocol Atomare Quantenbits, Rabi-Oszillationen 4

5 Atomares Quantenbit 2-Niveau-Atom Überlagerung Qubits in 40 Ca + ion Ground state + long-lived excited state Zeeman substates of level -1/2 1/2 5

6 Zeeman structure of the S 1/2 D 5/2 transition Zeeman structure in non-zero magnetic field: : D 5/2 5/2 3/2 1/2-1/2-3/2-5/2 2-level-system 1/2 5/2 S 1/2-1/2 1/2 Optical qubit transition Superpositions of S 1/2 (m=1/2) and D 5/2 (m=5/2) forms qubit P 3/2 Manipulation by laser pulses on 729 nm transition (~ 1 ms coherence time) P 1/2 τ 1 s 729 nm D 5/2 (m=5/2)> D 3/2 qubit S 1/2 S 1/2 (m=1/2)> 6

7 Qubit dynamics D> S> Rabi oscillations 7

8 Quantum bits: superpositions Laser excitation switches continuously between on and off Measurement shows either on (bright) or off (dark) % population (a) Time of excitation (µs) Rabi oscillations cont'd. 8

9 Das zentrale Werkzeug: Rabi-Oszillationen Resonant angeregtes 2-Niveau-Atom Absorption Stimulierte Emission Wahrsch. Zeit Laser Auf der betrachteten Zeitskala ist e> ein stabiler Zustand (zerfällt nicht spontan). Die Wahrscheinlichkeit, das Atom in g> oder e> zu finden, oszilliert mit Ramsey - Methode Resonant angeregtes 2-Niveau-Atom Absorption Stimulierte Emission Wahrsch. Zeit Laser "π/2 - Puls" "π/2 - Puls" Während der Laser aus ist, entwickelt sich die Quantenphase 9

10 Ramsey - Methode Resonant angeregtes 2-Niveau-Atom Absorption Stimulierte Emission Wahrsch. Atom in e> Zeit Laser "π/2 - Puls" "π/2 - Puls" Das Ergebnis des zweiten Pulses hängt von der Quantenphase ab. Ramsey - Methode Resonant angeregtes 2-Niveau-Atom Absorption Stimulierte Emission Wahrsch. Atom in g> Zeit Laser "π/2 - Puls" "π/2 - Puls" Das Ergebnis des zweiten Pulses hängt von der Quantenphase ab. 10

11 Ramsey - Methode Resonant angeregtes 2-Niveau-Atom Absorption Stimulierte Emission Wahrsch.?? Zeit Laser "π/2 - Puls" "π/2 - Puls" Sehr kleine Wechselwirkungen können die Quantenphase verändern. Ramsey - Methode Resonant angeregtes 2-Niveau-Atom Absorption Stimulierte Emission Wahrsch.?? Zeit Messung winziger Phasenverschiebungen 11

12 Ionenfalle und Bewegungs-Quantenbit Classic Paul trap endcap electrode z lens fluorescence detection ring electrode x y endcap electrode cooling beam 12

13 Ion storage Ion confinement requires a focusing force in 3 dimensions binding force, that is Quadrupole potential (saddle potential) Paul trap: (Penning trap: axial magn. field) Equation of motion in a Paul trap: This is a special case of the MATHIEU EQUATION (stability diagram ) Intuitive picture: time-averaged kinetic energy of driven motion = effective potential in which the ion oscillates freely with frequency Full motion = secular motion at with superimposed micromotion at (Intuitive picture works when ) Jan Huwer Doktorarbeit 13

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1D-solution of Mathieu equation single Al dust particle in trap position in trap secular motion at ω time

15 Trapping ions Paul trap mechanism Trapped charged particles Linear trap ~1 mm size Ion strings Laser cooling Localization << λ Laser Coulomb repulsion Interaction by common modes of motion Ion trajectory in a Paul trap 1D-solution of Mathieu equation single Al dust particle in trap position in trap secular motion at ω time micromotion at ξ Wuerker, Shelton, Langmuir, J. Appl. Phys. 30, 342 (1959) Secular motion = harmonic oscillator with frequency ω 15

16 Legacy : circular trap for single ions Miniature Paul trap Single atoms (Ba + ) in in trap 4.7 µm Ring Ø ~ 1 mm RF ~25 MHz, ~ 500V RF secular frequency ~ 1 MHz Linear ion trap evolution Paul mass filter Innsbruck Los Alamos München Boulder, Mainz, Aarhus Boulder 16

17 Innsbruck linear ion trap (2000) 1.0 mm 6 mm ω z MHz ω x, y MHz State-of-the-art linear Paul trap ( 40 Ca+ ions) R. Blatt ~100 µm 17

18 Ion strings in a linear Paul trap H.C. Nägerl et al., Appl. Phys. B 66, 603 (1998). First observations: Raizen et al., PRA 45, 6493 (1992), Waki et al., PRL 68, 2007 (1992). Quantized motion... 2> 1> n=0> 1> 0> motional qubit N ions 3 N oscillators 18

19 Motional sidebands 2-level-atom harmonic trap coupled system & transitions e... D 5/2 g Ω Γ ω { 2 1 n = 0 S 1/2 ω n> = 0> 1> 2> spectroscopy: carrier and sidebands n = 0 n = -1 n = 1 Laser detuning Rabi frequencies Carrier: Red SB: Blue SB: Ω Ω η n Ω η n +1 η = k <0 x 2 0> 1/2 «1 Excitation spectrum of the S 1/2 D 5/2 transition ω ax = 1.0 MHz ω rad = 5.0 MHz (only one Zeeman component) 19

20 Qubits in a single 40 Ca + ion internal qubit D 5/2 1> motional qubit... "computational subspace" D,0> D,1> 729 nm S 1/2 0> 2> 1> n=0> ω 1> 0> S,0> S,1> COHERENT LASER MANIPULATION (Rabi oscillations) S,n> D,n> : carrier transition ( = 0) S,n> D,n±1> : sideband transition ( = ±ω) First single-ion quantum gate: Monroe et al. (Wineland), PRL 75, 4714 (1995). Qubit rotations in computational subspace D,0> D,1> Laser-driven transitions are described by unitary operators (if Ω >> Γ D, Γ Laser ) : carrier: θ = Ω t C iφ + iφ ( ) θ R( θ, φ) = exp i 2 e σ + e σ S,0> S,1> red sideband: θ = Ω t SB iφ iφ ( ) θ + R1 ( θ, φ) = exp i 2 e σ a + e σ a computational subspace (levels with n>1 ignored) blue sideband: θ = Ω t + θ iφ + iφ R1 ( θ, φ) = exp i 2 ( e σ a + e σ a) SB where Example: excitation on blue sideband with 20

2 Ionen und quantenlogisches Gatter 2 ions + motion = 3 qubits With several ions, the motional qubits are shared motional qubit acts as the "bus" between the ions

21 2 Ionen und quantenlogisches Gatter 2 ions + motion = 3 qubits With several ions, the motional qubits are shared motional qubit acts as the "bus" between the ions vibrational modes computational subspace: 2 ions, 1 mode D,D,1> D,D,0> laser on ion 2 laser on ion 1 D,S,1> D,S,0> S,D,1> S,D,0> laser on ion 1 laser on ion 2 S,S,1> S,S,0> 21

22 Excitation spectrum of two ions Quantum gate proposal(s) controlled NOT ε 1 ε 2 ε1 ε1 ε control bit bit target bit bit Further gate proposals: Cirac & Zoller Mølmer & Sørensen, Milburn Jonathan & Plenio & Knight Geometric phases 22

Quantum gate proposal(s) phase gate 0 0 1 1 0 1 0 1 0 0 0 1 1 0 1 1 control bit bit target bit bit Further gate proposals:

23 Quantum gate proposal(s) phase gate control bit bit target bit bit Further gate proposals: Cirac & Zoller Mølmer & Sørensen, Milburn Jonathan & Plenio & Knight Geometric phases Details of C-Z gate operation (Phase gate) 23

24 Desired result, schematic S S S D D S D D S S S D D D D S control target Cirac-Zoller two-ion controlled-not gate ε1 ε 2 Preparation Detection S> = bright D> = dark ion 1 motion ion 2 S, D 0 0 S, D SWAP CNOT SWAP -1 control qubit "bus" qubit target qubit 24

25 ion 1 motion ion 2 Cirac-Zoller two-ion controlled-not operation S S Ion 1,, D D pulse sequence Ion 2 SWAP SWAP blue π 0 c π/2 0 blue π/2 ½ 0 CNOT Phase blue π π/2 blue π/2 ½ 0 blue π π/2 c π/2 π blue π π control bit bit target bit bit CNOT gate = Phase gate enclosed by π/2- pulses Phase gate implemented by composite pulses Photonenrückstoß und Laser-Kühlung 25

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27 = 27

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29 Coherent state manipulation D,0 D,1 carrier S,0 S,1 carrier and sideband Rabi oscillations with Rabi frequencies sideband Lamb-Dicke parameter Each point : average of individual measurements, preparation coherent rotation state detection 29

30 e,0> e,1> e,2> g,0> g,1> g,2> Laser cooling of trapped atoms Photon recoil: p> p +ħk laser +ħk spont > laser spont. In trap: transitions between energy eigenstates n> n'> atom motion Lamb-Dicke regime: only n> n±1> Cooling principle: cooling transitions more probable than heating transitions Final temperature: equilibrium of heating and cooling 30

31 Doppler and sideband cooling Γ > Ω trap Doppler cooling Ω Absorption Γ/2 Γ < Ω trap Absorption ν laser ν atom ν Sideband cooling Ω Γ leads into ground state! Ground state cooling with Γ > Ω? Narrow resonances through quantum interference in multilevel system ν laser ν atom ν Methoden 31

32 Ion trap setup RF linear trap: ω 0.7-2Mhz ω 5MHz Longitudinal confinement Radial confinement: oscillating saddle potential Double trap apparatus 32

Innsbruck linear ion trap GND RF RF 5mm GND +HV +HV ω axial ω radial 0.

$), Φ, Ampl RF AOM AOM = acousto-optical acousto-opticalmodulator, based basedon on Bragg Braggdiffraction$

33 Innsbruck linear ion trap GND RF RF 5mm GND +HV +HV ω axial ω radial MHz 5 MHz Two 2-level systems 1> 0> 1> 0> Laser pulses for coherent manipulation etc cw laser τ coh >> τ gate I ν t AOM ν+, Φ, Ampl I Φ 2 Φ 1 Φ 3 to trap t (That's the difficult bit!), Φ, Ampl RF AOM AOM = acousto-optical acousto-opticalmodulator, based basedon on Bragg Braggdiffraction "Ampl" "Ampl" = Amplitude, Amplitude, includes includesswitching switchingon/off 33

Addressing of ions in a string Well-focussed laser beam beam steering with electro-optical deflector addressing waist ~ 2.5-3.0 mm < 1/400 intensity on neighbouring ion first demonstration: H.C.

34 Addressing of ions in a string Well-focussed laser beam beam steering with electro-optical deflector addressing waist ~ mm < 1/400 intensity on neighbouring ion first demonstration: H.C. Nägerl et al., Phys. Rev. A 60, 145 (1999) Discrimination of qubit states State detection by photon scattering on S 1/2 to P 1/2 transition at 397 nm P 3/2 τ 8 ns Photons observed : S 1/2 = S> No photons : D 5/2 = D> P 1/2 397 nm 866 nm τ 1 s D 5/2 D> D 3/2 qubit Detector S 1/2 S 1/2 S> 34

State detection: shelving P monitor S D (Shelving level can, but need not be

2 1 Histogram of counts in 9 ms Poisson distribution N ± N ½ discrimination

85% D-Zustand D state occupied besetzt S S-Zustand state occupied besetzt 0

Dehmelt 1975 Quantum state discrimination with 2 ions Individual ion

35 State detection: shelving P monitor S D (Shelving level can, but need not be the same as qubit state) Anzahl # of measurements der Messungen Histogram of counts in 9 ms Poisson distribution N ± N ½ discrimination efficiency 99.85% D-Zustand D state occupied besetzt S S-Zustand state occupied besetzt Zählrate pro 9 ms counts in 9 ms H. Dehmelt 1975 Quantum state discrimination with 2 ions Individual ion detection on CCD camera Two-ion histogram (1000 experiments) 5µm SS> DS> SS> region 1 region 2 SD> DS> DD> SD> DD> quantum state populations p SS,p SD,p DS,p DD 35

out of CS! π 2 Ω Rabi ~η 2 computational subspace D,0 π S,0 π D,1 S,1 4π A.M. Childs et al., Phys. Rev.

36 Gate pulses (I) : SWAP Swap information from internal into motional qubit and back naive idea : π-pulse on blue SB (works if initial state is not S,1>) composite SWAP (from NMR) computational subspace D,0 S,0 π D,1 S,1 Ω Rabi ~η 1 out of CS! π 2 Ω Rabi ~η 2 computational subspace D,0 π S,0 π D,1 S,1 4π A.M. Childs et al., Phys. Rev. A 63, (2001) 3-step composite SWAP operation π on D,1 S, 2 π on D, 0 S,1 I. Chuang et al., Innsbruck (2002) 36

37 Hiding qubits Detect quantum state of one ion only (needed for teleportation) Protect neighbours from addressing errors D 5/2 D 5/2 π S 1/2 S 1/2 ion #1 ion #2 D D D 5/2 D 5/2 S 1/2 S 1/2 ion #1 ion #2 superposition state of ion #2 protected Teleportation eines atomaren Zustands 37

38 38

39 Teleportation protocol Teleportation in Innsbruck (Riebe et al.) "Hide": protect ion from interaction with laser pulses (manipulation or measurement) on the other ions 39

40 Quantum teleportation with atoms: result 83 % class.: 67 % no cond. op.: 50 % M. Riebe et al., Nature 429, 734 (2004) 40

Motion and motional qubit

Quantized motion Motion and motional qubit... > > n=> > > motional qubit N ions 3 N oscillators Motional sidebands Excitation spectrum of the S / transition -level-atom harmonic trap coupled system & transitions