Quantum computation with trapped ions

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Laura Blake
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(electronic), while laser cooling has evolved into the control of the external degree of freedom, i.e. of the motional quantum of the ion in the trap.

1 Abstract Since the first preparation of a single trapped, laser-cooled ion by Neuhauser et el. in 198, a continuously increasing degree of control over the of single ions has been achieved, such that what used to be spectroscopy has turned into coherent manipulation of the internal (electronic), while laser cooling has evolved into the control of the external degree of freedom, i.e. of the motional quantum of the ion in the trap. Based on these developments, Cirac and Zoller proposed in 1995 to use a trapped ion string for processing quantum information, and they described the operations required to realise a universal two-ion quantum logical gate. This seminal proposal sparked intense experimental activities in many groups and has led to spectacular results. In this lecture, the basic experimental techniques which enable quantum information processing with trapped ions and atoms will be reviewed. In particular, it will be explained how the fundamental concepts of quantum computing, such as quantum bits (qubits), qubit rotations, and quantum gates, translate into experimental procedures in a quantum optics laboratory. Furthermore, the recent progress of quantum computing with ions and atoms will be summarised, and some of the new approaches to meet future challenges will be mentioned. It is intended to provide an intuitive understanding of the matter that should enable the non-specialist student to appreciate the paradigmatic role and the potential of trapped single ions and atoms in the field of quantum computation. Quantum computation with trapped ions Fundamentals ion ion & atom traps, quantum bits, quantum gates Implementations -qubit gates, teleportation, recent progress Scaling up: QC architecture Quantum networks: atom-photon interfaces Jürgen Eschner ICFO The Institute of Photonic Sciences Castelldefels (Barcelona), Spain Les Houches School of Physics, Singapore, 4 July, 9 Towards quantum technology (Moore's Law) Quantum information with single quantum systems ENIAC (1947) Pentium 4 () 1 atom 1 atom per bit How many atoms per bit of information? W. R. Keyes, IBM J. Res. Dev. 3, 4 (1988) & C. Schuck, ICFO ~ Quantum effects will play a role and open up new avenues Applications in informatics and physics P. Shor, 1994: factorization of large numbers is polynomial on a quantum computer, exponential on a classical computer L. Grover, 1997: data base search N 1/ quantum queries, N classical simulation of Schrödinger equations or any unitary evolution quantum cryptography / repeaters / quantum links improved atomic clocks (P. Schmidt et al., 5) (Gedanken-)experiments fundamentals of QM, decoherence, entanglement Fundamental phenomena Quantum information Technological applications 1

Various approaches Single trapped and laser-cooled ions... Ion traps Neutral atoms in traps (opt. traps, opt.

.. are cold (< 1 mk) and well-localized (< nm) Ł Application for precision measurements and frequency standards (work by NIST, PTB, MPQ, NRC, NPL, NML, Mainz, MIT,.

) Optical qubits and LOQC (linear optics quantum computation) Electrons on L-He surfaces Electrons in Penning trap Spectral hole burning in ion-doped crystals Colour

.. are individual quantum systems Ł Textbook experiments on fundamental quantum mechanics (work by NIST, Innsbruck, MPQ, Seattle, IBM, Hamburg,...).

2 Various approaches Single trapped and laser-cooled ions... Ion traps Neutral atoms in traps (opt. traps, opt. lattices, microtraps) Neutral atoms and cavity QED NMR (in liquids)... are cold (< 1 mk) and well-localized (< nm) Ł Application for precision measurements and frequency standards (work by NIST, PTB, MPQ, NRC, NPL, NML, Mainz, MIT,...) Superconducting qubits (charge-, flux-qubits) Solid concepts (spin systems, quantum dots, etc.) Optical qubits and LOQC (linear optics quantum computation) Electrons on L-He surfaces Electrons in Penning trap Spectral hole burning in ion-doped crystals Colour centers and more COLD SYSTEMS nm... are individual quantum systems Ł Textbook experiments on fundamental quantum mechanics (work by NIST, Innsbruck, MPQ, Seattle, IBM, Hamburg,...) allow allow full full control, control, manipulation manipulation and and measurement measurement of of quantum quantum ( quantum ( quantum engineering ) engineering ) Ł Application for studies of entanglement and decoherence Ł Application for quantum information processing and and for quantum information processing studies of entanglement and decoherence (work by NIST, Innsbruck, Hamburg, Oxford, MPQ, Michigan, ICFO) (work by NIST, Hamburg, Oxford, MPQ, Ibk,...) Classic Paul trap endcap electrode z lens fluorescence detection Ion traps ring electrode x y endcap electrode cooling beam

Ion storage Ion trajectory in a Paul trap Ion confinement requires a focusing force in 3 dimensions r r r r r r

Quadrupole potential Paul trap: Φ = U + V cosωt Φ Φ = ( x + y z ) r (Penning trap: Φ = U + axial magn.

a special case of the MATHIEU EQUATION (stability diagram ) Intuitive picture: time-averaged kinetic energy of driven

(Intuitive picture works when ) ω, ω, ω Ω x y ω << Ω with superimposed micromotion at z secular motion at ω time

3, 34 (1959) Secular motion = harmonic oscillator with frequency ω Legacy : circular trap for single ions Linear ion

3 Ion storage Ion trajectory in a Paul trap Ion confinement requires a focusing force in 3 dimensions r r r r r r binding force F ~, that is F = ee = e Φ Φ ~ 1D-solution of Mathieu equation single Al dust particle in trap Quadrupole potential Paul trap: Φ = U + V cosωt Φ Φ = ( x + y z ) r (Penning trap: Φ = U + axial magn. field) m & e (saddle potential) Equation of motion in a Paul trap: x + ( U + V cos Ωt) x = r position in 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 ω Full motion = secular motion at (Intuitive picture works when ) ω, ω, ω Ω x y ω << Ω with superimposed micromotion at z secular motion at ω time micromotion at ξ Wuerker, Shelton, Langmuir, J. Appl. Phys. 3, 34 (1959) Secular motion = harmonic oscillator with frequency ω Legacy : circular trap for single ions Linear ion trap evolution Miniature Paul trap Single atoms (Ba + ) in in trap Paul mass filter 4.7 µm Innsbruck Los Alamos Ring Ø ~ 1 mm RF ~5 MHz, ~ 5V RF secular frequency ~ 1 MHz Boulder, Mainz, Aarhus Boulder München 3

The prototype State-of-the-art linear Paul trap ( 4 Ca+ ions) Ions = "qubits" = q.m.

measured Cirac & Zoller, PRL 1995 R. Blatt H.C. Nägerl et al., Appl. Phys.

TLS) Encoding of quantum information requires longlived atomic s: Single quantum systems - Qubits

alkalis: Ca +, Sr +, Ba +, Ra +, (Yb +, Hg + ) etc.

4 The prototype State-of-the-art linear Paul trap ( 4 Ca+ ions) Ions = "qubits" = q.m. coherent -level systems Laser-controlled Interacting through vibrational modes Individually measured Cirac & Zoller, PRL 1995 R. Blatt H.C. Nägerl et al., Appl. Phys. B 66, 63 (1998); first ion strings: NIST, MPQ ~1 µm Qubits in trapped ions (two-level systems, TLS) Encoding of quantum information requires longlived atomic s: Single quantum systems - Qubits optical transitions (forbidden transitions, intercombination lines) S D transitions in earth alkalis: Ca +, Sr +, Ba +, Ra +, (Yb +, Hg + ) etc. Innsbruck microwave transitions (Hyperfine transitions, Zeeman transitions) earth alkalis: 9 Be +, 5 Mg +, 43 Ca +, 87 Sr +, NIST 137 Ba +, 111 Cd +, 171 Yb + Examples taken from: TLS TLS 4

Level scheme of 4 Ca + Zeeman structure of the transition 393 nm τ 8 ns 397 nm 854 nm 866 nm τ 1 s Zeeman structure in non-zero magnetic field: : 5/ 3/ 1/ - 1/ -3/ -5/ -level-system 1/ 5/ 79 nm

5 Level scheme of 4 Ca + Zeeman structure of the transition 393 nm τ 8 ns 397 nm 854 nm 866 nm τ 1 s Zeeman structure in non-zero magnetic field: : 5/ 3/ 1/ - 1/ -3/ -5/ -level-system 1/ 5/ 79 nm (electric quadrupole) - 1/ 1/ Qubit transition Qubit dynamics Superpositions of (m=1/) and (m=5/) forms qubit Manipulation by laser pulses on 79 nm transition (~ 1 ms coherence time) qubit τ 1 s (m=5/)> D> 79 nm (m=1/)> S> 5

6 Rabi oscillations Rabi oscillations cont'd. Bloch sphere Discrimination of qubit s State detection by photon scattering on to transition at 397 nm τ 8 ns Photons observed : = S> No photons : = D> 397 nm 866 nm τ 1 s D> qubit Detector S> 6

State detection: shelving Quantum jumps & "Quantum amplification" P monitor S D (Shelving level can, but need not be the same as qubit ) Anzahl # of measurements der Messungen 8 D-Zustand D occupied

7 State detection: shelving Quantum jumps & "Quantum amplification" P monitor S D (Shelving level can, but need not be the same as qubit ) Anzahl # of measurements der Messungen 8 D-Zustand D occupied besetzt Histogram of counts in 9 ms Poisson distribution N ± N ½ discrimination efficiency 99.85% S S-Zustand occupied besetzt Zählrate counts pro in 99 ms ms 397 nm 393 nm 854 nm 866 nm 85 nm 4 Ca + 1 sec 79 nm High power LED for photo ionisation (8 385 nm, HWHM 15 nm) excites 393 nm transition Ions are pumped into (life time ~ 1s) No fluorescence H. Dehmelt 1975 Detection of absorption of < 1 photon/sec F. Rohde, C. Schuck, M. Almendros, M. Hennrich, J.E. 7

Quantum information processing with trapped atoms

Quantum information processing with trapped atoms Introduction Fundamentals: ion iontraps, quantum bits, bits, quantum gates Implementations: 2-qubit gates, teleportation, More recent, more advanced, Jürgen