1. Gregynog QIP meeting QIP Experiments with ions, atoms and molecules Christopher Foot, University of Oxford c.foot@physics.ox.ac.uk

2. Objectives of these lectures: Describe QIP experiments that use atomic and molecular physics (not including NMR) General requirements for a quantum gate between two qubits – spin-dependent interaction Review experimental techniques and some current experiments with ion in traps Neutral atoms in optical lattices: simulation of condensed matter physics and QIP applications Recent ideas for QIP with polar molecules = hybrid of atoms and ions

3. Quantum gate - controlled operation CNOT gate: CROT gate (or controlled-Z gate) Exercise: Show how to construct a CNOT gate from a controlled-Z gate and 2 Hadamard gates. (Ex 4.17 in Nielson & Chuang)

4. CROT gate (or controlled-Z gate) based on state-dependent (spin-dependent) interaction or Interaction 0 0 11 where Qubit AQubit B

5. Comment on quantum gates Equivalent to Difficult to implement since requires very precise control of experimental timing.

6. `Pushing gate– laser beam exerts state-dependent force on ions Harmonic trapping potential Repulsive Coulomb force 1 0 E Phase factor Phase difference Cf. `wobble gate discussed later Repulsive Attractive

7. Summary of Lecture 1: Ions 1.Requirements for QIP 2.Ion trap principles 3.Read out and Quantum jumps 4.Manipulation of single qubit by Raman transitions 5.Laser cooling to the lowest vibrational level 6.Current experimental capability in Oxford and elsewhere 7. Survey of ideas being actively explored in experimental groups 8.How to make a computer,

8. Requirements for quantum computing Excellent quality qubits Q = T coherence / T gate = 10 6 now realised for ions move qubits around quickly and without error High precision gate between neighbours (error = 10 -4 ) Single-qubit gates Measurement of qubits (read out) D. Deutsch, Proc R Soc Lond A 400, 97 (1985) & 425, 73 (1989). A.Steane, Phys. Rev. A 68, 042322 (2003) & Quant. Inf. Comp. 2, 297 (2002). *

9. Basic ion trap methods N.B. Most of the slides in this lecture come from the Ion trapping group in Oxford (part of the IRC)

10. Paul Trap r.f. quadrupole + end caps = Axial confinement by electrostatic (quadratic) potential. Radial confinement by oscillating quadrupole potential. Electrostatic trapping not possible, see Foot, Atomic Physics, OUP 2005

11. The trap Axial motional freq. of order MHz Ion-electrode distance = 0.1 to 1 mm 7 mm

12. Alkali-like ions Choice of ion: Want simple energy level structure when singly-ionised Group II or other metals: X = Be, Ca, Sr, Ba, Yb, Cd, Hg Ground configuration of X + ion has electron spin s = ½. Hyperfine structure arising from interaction of nuclear spin (nuclear magnetic moment) with magnetic field created by the unpaired electron.

13. The hyperfine structure of Ca-43, the ion used in Oxford experiments, is inverted B field-independent 1 st order Zeeman sensitive Use transition with no first-order Zeeman effect, which is therefore insensitive to magnetic field fluctuations, cf. atomic clocks. Qubit coherence time of order seconds (see later)

14. Readout by fluorescence on cycling transition 2 S 1/2 F=1, M F =1 F=2, M F =2 1.25 GHz 2 P 3/2 2 P 1/2 F=3, M F =3 E.g. Be+ Selection rules cycling transition PM tube or CCD camera

15. Electron shelving or quantum jumps

16. Time needed to measure a qubit at 99.9% fidelity Collection & photon detection efficiency = 0.02 excited state lifetime = 5 ns required photon count for P(error) < 0.001 is 10 photons time to count 10 photons = 2 x 10 / = 5 s Current experiments allow 100 s to 1 ms Poissonian distribution of photon count Scattering rate

17. The trap Excellent signal-to-noise in detection of individual ions 7 mm Comment: There is well-developed and efficient scheme for reading out the state of ions. This is not yet achieved for neutral atoms or molecules.

18. Single bit rotations Microwaves: U on all qubits at once Stimulated Raman transition: U on a chosen individual qubit 1-bit gate time ' 1 micro-second

19. Raman transition 2 S 1/2 F=1, M F =1 F=2, M F =2 1.25 GHz hyperfine structure 313 nm 2 P 3/2 2 P 1/2 AOM: shift 1.25 GHZ Very high-precision and stable phase

20. Raman transition: effective 2-level system 1 2 100 GHz 500 MHz 10 MHz photon scattering = 10 -4 = 1.25 MHz (unwanted) photon scattering rate

21. 2 S 1/2 F=1, M F =1 F=2, M F =2 2 P 3/2 2 P 1/2 Qubit decoherence per gate time Photons scattered per gate time PRL 95, 030403 (2005)

22. Trapped atom: quantum simple harmonic motion z z 1 2 0 z 1 2 0 0 0

23. Laser cooling of trapped ions: Sideband cooling 3 2 0 L 0 z 1 2 0 1 3 << 1 Resultant thermal distribution: Laser cooling a trapped ion very different to cooling of free atoms

24. Measure temperature of trapped ion 1 0 1 2 0 2 PgPg L Compare excitation probability of first red sideband and first blue sideband 0 Cf Fig. 12.10 in Foot (2005)

25. Laser cooling of trapped ions: Sideband cooling 3 2 0 L 0 z 1 2 0 1 3 Experimental results: 2005 Oxford 0.02 << 1

26. Current experimental capability in Oxford Dr David Lucas Prof. Andrew Steane Prof. Derek Stacey ……………. N.B. Most of the slides in this lecture come from the Ion trapping group in Oxford (part of the IRC)

27. Main results VERY LONG COHERENCE DETERMINISTIC ENTANGLEMENT OF ION SPIN QUBITS

28. Very long (1s) qubit coherence time field-independent, prepare with ~15% efficiency 1 st order Zeeman sensitive, prepare with 100% efficiency Readout by shelving with 95% efficiency Rabi flopping and Ramsey experiments using 3.2 GHz microwaves Ca-43 hyperfine qubit

29. Rabi flopping microwave pulse length (ms) data fit P(F=4,M F =0)

30. Coherence (T 2 ) time control experiment (small delay) Ramsey experiment (long delay) Detuning (Hz) from 3,225,611,696 Fringe visibility vs. delay Ramsey fringes = 0.8(2) s Ramsey gap (ms)

32. Exercise Show why the pulse sequence /2 - - /2 is more robust than a Ramsey experiment ( /2 - /2 sequence) Also called Square root of NOT gate Cf Exercise 7.3 in Foot, Atomic Physics NOT gate:

33. Entanglement

34. Spin-dependent oscillating force Dipole force in standing wave B 50 m Raman beams F Laser standing wave produces an oscillating force on a pair of ions Robust and fast To be described in detail later + ( Explain how state-dependence of the force arises in next lecture. Force/potential depends on the polarization of the light. ) *

35. Leibfried or wobble gate D. Leibfried et. al. Nature 422, 412 (2003) Equivalent to controlled-not

36. Really nice gate 1.Only the area, not the shape of the loop matters 2.Non-zero ion temperature? –just displace the starting point, –same loop no problem 3.Shift of laser standing wave phase? –rotate the loop about the initial point –same loop no problem

37. Deterministic entanglement by phase gate spin qubit ( 40 Ca ground state) stretch mode freq. 866 kHz ion separation 9 m (= 22 = 0.35, < 0.1 gate time 77 s ( s = 67) fidelity 75 (5) % cool | 2, ) 2 measure 2 analysis pulseprepare gate

38. Wobble gate results July 2005 data: use twin loop, fidelity 82(2)% limited largely by photon scattering ( = 30 GHz) and laser intensity noise D. Lucas, M. McDonnell, S. Webster, J. Home, B. Keitch, D. Stacey, A. Ramos, A. Steane

39. Tomography Deduced density matrix hence entanglement of formation E = 0.52

40. Two-qubit gates Use laser-driven oscillatory motion of ions: Current experiments (2 to 8 ions): –fidelity » 90% (97% reported) –gate time » 10 to 100 s Future: –fidelity 99.99% (10 -4 ) with good (bright and stable) lasers –time 100 ns to 1 s

41. Requirements for quantum computing Excellent quality qubits Q = T coherence / T gate = 10 6 now realised for ions move qubits around quickly and without error High precision gate between neighbours (error = 10 -4 ) Single-qubit gates Measurement of qubits (read out) D. Deutsch, Proc R Soc Lond A 400, 97 (1985) & 425, 73 (1989). A.Steane, Phys. Rev. A 68, 042322 (2003) & Quant. Inf. Comp. 2, 297 (2002).

42. Moving information around the machine m Logical information encoded in large groups of ions. QEC uses a lot of parallel ops. (Animated version, Chuang website.) Kielpinski, Monroe, Wineland, Nature 417,709 (2002)

43. Moving ions around -Already mentioned by Prof Knight

44. 7-zone trap, Oxford/Liverpool collaboration ion-electrode distance = 0.7 mm trap-trap separation = 0.8 mm test open design concept Built by University of Liverpool, S. Taylor

45. Moving quantum information around Array of ion traps with ions transported between traps Possibly use large numbers of ions in same trap Possibly use ionphoton coupling

46. Ion entangled with photon (2004) March 2004 Blinov, Moehring, Duan and Monroe

47. Scalable computer by cluster methods Duan et al, Quant. Inf. Comp. 4, 165 (2004)

48. multi-qubit controlled entanglement 4,5,6- ion cat state |0000 + |1111 F > 0.76, W< 0.51 |00000 + |11111 F > 0.60, W< 0.2 |000000 + |111111 F > 0.51, W< 0.02 NIST, Boulder, USA 3 to 8- ion W state |001 + |010 + |100 F = 0.82, W= 0.53 |0001 + |0010 + |0100 + |1000 F = 0.60, W= 0.46... |00000001 + |00000010 + F = 0.72, W= 0.029 Innsbruck, Austria

49. Requirements for quantum computing Excellent quality qubits Q = T coherence / T gate = 10 6 now realised for ions move qubits around quickly and without error High precision gate between neighbours (error = 10 -4 ) Single-qubit gates Measurement of qubits (read out) Conclusion: Ion trapping fulfils these requirements. No obvious `roadblock in the way of QIP with ions, `just requires development of technology

50. How to make a computer quant-ph 2004