1. Vorlesung Quantum Computing SS 08 1 A scalable system with well characterized qubits Long relevant decoherence times, much longer than the gate operation time A qubit-specific measurement capability A A universal set of quantum gates U The ability to initialize the state of the qubits to a simple fiducial state, e.g. |00...0> DiVincenzo criteria DiVincenzo: Fortschr. Phys. 48 (2000) 9-11, pp. 771-783

2. Vorlesung Quantum Computing SS 08 2 Quantum Computing with Ions in Traps How to trap ions State preparation Qubit operations CNOT Deutsch – Jozsa Algorithm advantages/drawbacks

3. Vorlesung Quantum Computing SS 08 3 Paul Trap Nobel Prize 1989 centre is field free quadrupole field x and y motions not coupled! Chemnitz University

4. Vorlesung Quantum Computing SS 08 4 Linear Trap x y z U1U1 R U ac U ac (t) = U r + V 0 cos T t effective potential: eff = x 2 x 2 + y 2 y 2 + z 2 z 2 x = y >> z (averaged over one rf cycle) U2U2 z0z0 M. Sasura and V. Buzek: quant-ph/0112041

5. Vorlesung Quantum Computing SS 08 5 Potential

6. Vorlesung Quantum Computing SS 08 6 Ions in a Linear Trap z = 2 qU 12 mz 0 2 typical operation parameters: V 0 = 300 – 800 V T /2 = 16 – 18 MHz U 12 = 2000 V z 0 = 5 mm R = 1.2 mm z /2 = 500 - 700 kHz x,y /2 = 1.4 – 2 MHz ( 40 Ca + ) 70 m 40 Ca + 24 Mg + Seidelin et al: Phys. Rev. Lett. 96, 253003 (2006) Nägerl et al: Phys. Rev. A 61, 023405 (2000)

7. Vorlesung Quantum Computing SS 08 7 quantum computing with ions HH -1 calculation U preparation read-out |A| time the ions are prepared to be in their ground state Doppler coolingside band cooling 1st step2nd step k B T << ħ z

8. Vorlesung Quantum Computing SS 08 8 Doppler cooling when absorbing a photon, also the momentum is transferred the net momentum of the spontaneous emission is zero E = ħ p = ħk E = 0 p = 0 E = ħ p = ħk k absorption for ions moving toward the laser beam the light appears blue shifted use a red detuned laser = 0 + k v Ca

9. Vorlesung Quantum Computing SS 08 9 side band cooling Doppler cooling gets down to k B T ħ internal electronic ground and excited state |g,|e trapped ions moving in harmonic potential states |n, n= 0,1,2… cooling: |g,n |e,n-1 |e,n-1 |g,n-1

10. Vorlesung Quantum Computing SS 08 10 ions used as qubits electronic states as qubits (pseudo-spin) (CNOT, Deutsch-Jozsa Algorithm, Quantum-Byte) hyperfine states as qubits (CNOT, error correction, Grover Algorithm)

11. Vorlesung Quantum Computing SS 08 11 40 Ca + as qubit 4 2 S 1/2 4 2 P 1/2 4 2 P 3/2 3 2 D 3/2 3 2 D 5/2 397 nm 729 nm 854 nm 866 nm |0 |1 quadrupole transition used for Laser cooling quadrupole transition with relatively long relaxation time for cooling: 866 nm transition has to be irradiated as well, otherwise charge carriers will be trapped in 3 2 D 3/2 orbital fluorescence detection for read-out detection Nägerl et al: Phys. Rev. A 61, 023405 (2000) D 5/2 occupation P D red sideband blue sideband after Doppler cooling

12. Vorlesung Quantum Computing SS 08 12 9 Be + as qubit electron spin S = 1/2, m s = 1/2 nuclear spin I = 3/2, m I = 1/2, 3/2 F = I + S, m F 2 2 P 3/2 2 2 P 1/2 12 GHz 2 2 S 1/2 |F=2, m F =2 |F=1, m F =1 | | 1.25 GHz Doppler cooling hyperfine levels have long relaxation times sideband cooling |2,2 |n |1,1 |n-1 ; induced spontaneous Raman transition |1,1 |n-1 |2,2 |n-1 sideband cooling detection with fluorescence after + excitation + detection Monroe et al: Phys. Rev. Lett. 75, 4011 (1995) red blue after Doppler cooling after sideband cooling

13. Vorlesung Quantum Computing SS 08 13 quantum computing HH -1 calculation U preparation read-out |A| time quantum-bit (qubit) 0 1 a 1 0 + a 2 1 = a1a1 a2a2

14. Vorlesung Quantum Computing SS 08 14 qubit operations how does the system evolve with time? U (t) e ħ - iH QC t ^ ^ H QC = H trap + H ion + H man ^^^^ Splitting of S = ½ in external magnetic field: 2 2 S 1/2 |F=2, m F =2 |F=1, m F =1 | | s /2 = 1.25 GHz H ion = - ħ s 2 1 0 0 ^ B 0 = 0.18 mT H ion = - SB = - S z B 0 = - L S z ^ ^^

15. Vorlesung Quantum Computing SS 08 15 qubit coupling coulomb repulsion couples motional degrees of freedom H trap = ( x 2 x i 2 + y 2 y i 2 + z 2 z i 2 + ) + M 2 pi2pi2 M2M2 e2e2 4 0 |r i - r j | i=1 N N j>i trap potential eff E kin coulomb potential positions at rest 1 mode 2 modes A. Steane: quant-ph/9608011 ^

16. Vorlesung Quantum Computing SS 08 16 vibration modes as qubits (bus) centre of mass motion used as qubit A. Steane: quant-ph/9608011 i=1 H trap = ( z 2 z i 2 + ) = ħ i a i a i pi2pi2 M2M2 M 2 NN i z = 2 qU 12 mz 0 2 z z 3 J.F. Poyatos et al., Fortschr. Phys. 48, 785

17. Vorlesung Quantum Computing SS 08 17 9 Be + : the two qubit system 2 2 P 1/2 50 GHz 2 2 S 1/2 |F=2, m F =2 |F=1, m F =1 | | |1 | |0 |1 | |0 s /2 = 1.25 GHz z /2 = 11.2 MHz 2 2 P 3/2 |F=3, m F =3 |0 |aux |F=2, m F =0 vibrational state: control qubit hyperfine state: target qubit Raman transition + detection ~ 313 nm

18. Vorlesung Quantum Computing SS 08 18 spin dynamics dM x dt = (M y (t)B z M z (t)B y ) dM y dt = (M z (t)B x M x (t)B z ) dM z dt = (M x (t)B y M y (t)B x ) = M y (t)B z = - M x (t)B z = dM dt = M(t) x B = M y cos( L t) - M x sin( L t) = M x cos( L t) + M y sin( L t) B = 0 0 BzBz B 1 cos t B 1 sin t B0B0 magnetic field rotating in x,y-plane

19. Vorlesung Quantum Computing SS 08 19 spin flipping in lab frame http://www.wsi.tu-muenchen.de/E25/members/HansHuebl/animations.htm

20. Vorlesung Quantum Computing SS 08 20 rotating frame x y z x y z cos t sin t - sin t 001 0 0 = r z y x xrxr yryr t t cos t sin t - sin t 001 0 0 cos t sin t 0 B1B1 cos t -sin t 0 B1B1 + B rf = r cos 2 t 0 B rf = r 1 0 0 B1B1 -sin 2 t B1B1 + constant counter-rotating at twice RF applied RF generates a circularly polarized RF field, which is static in the rotating frame B 1 cos t 0 0 2 =

21. Vorlesung Quantum Computing SS 08 21 spin flip in rotating frame http://www.wsi.tu-muenchen.de/E25/members/HansHuebl/animations.htm

22. Vorlesung Quantum Computing SS 08 22 qubit manipulation: laser interaction H man = - B = m S B = B 1 x cos(kz- t+ ) ^ H man (S + e i + S - e -i ) = m B 1 /2ħ ħ 2 frame of reference: H 0 = ħ s S z + ħ z a a only spin state is changed i (S + ae i - S - a e -i ) ħ 2 for = s - z red side band i (S + a e i - S - ae -i ) ħ 2 for = s + z blue side band change of vibrational state always implies change of spin state Lamb-Dicke parameter: 2 d 0 / << for = s

23. Vorlesung Quantum Computing SS 08 23 qubit rotation 2 2 P 1/2 50 GHz 2 2 S 1/2 |F=2, m F =2 |F=1, m F =1 | | |1 | |0 |1 | |0 s /2 |0 |aux |F=2, m F =0 Qubit rotation on target qubit U /2,ion Raman transition with detuning s Duration of laser pulse: /2 rotation 2 e = i ħ S y cos /4 sin /4 - sin /4 1 1 1 1 2 = = U /2

24. Vorlesung Quantum Computing SS 08 24 /2- rotation matrix - - 1 2 U /2,ion = base vectors of the two–qubit register: Transformation matrix: U /2,ion = ( ) 1 2 1 2 1 2 1 2

25. Vorlesung Quantum Computing SS 08 25 CNOT operation - U ph = U ph transformation matrix: Transformation sequence: U /2,ion =U - /2,ion U ph - - 1 2 - - - = = U CNOT Monroe et al: Phys. Rev. Lett. 75, 4714 (1995)

26. Vorlesung Quantum Computing SS 08 26 phase rotation 2 2 P 1/2 50 GHz 2 2 S 1/2 |F=2, m F =2 |F=1, m F =1 | | |1 | |0 |1 | |0 s /2 |0 |aux |F=2, m F =0 Phase rotation on control qubit U ph Raman transition between and auxiliary state Full rotation by 2 U ph

27. Vorlesung Quantum Computing SS 08 27 quantum computing HH -1 calculation U preparation read-out |A| time quantum-bit (qubit) 0 1 a 1 0 + a 2 1 = a1a1 a2a2

28. Vorlesung Quantum Computing SS 08 28 Be ions: read-out spin state |F=2, m F =2 | |1 | |0 2 2 P 3/2 |F=3, m F =3 2 2 S 1/2 |F=1, m F =1 | |1 | |0 + detection read-out spin state via fluorescence prepare desired initial state using Raman pulses | on blue side band |1 on internal state |1 | perform CNOT cool system to | |0

29. Vorlesung Quantum Computing SS 08 29 Be ions: read out vibrational state |F=2, m F =2 | |1 | |0 2 2 P 3/2 |F=3, m F =3 2 2 S 1/2 |F=1, m F =1 | |1 | |0 + detection read-out spin state prepare same initial state and do CNOT convert vibrational into spin state on red side band for | on blue side band for | read-out spin state via fluorescence