Quantum computation with trapped ions

Jürgen Eschner

ICFO - Institut de Ciencies Fotoniques, 08860 Castelldefels (Barcelona), Spain

Summary. — The development, status, and challenges of ion trap quantum com-puting are reviewed at a tutorial level for non-specialists. Particular emphasis is puton explaining the experimental implementation of the most fundamental conceptsand operations, quantum bits, single-qubit rotations and two-qubit gates.

PACS 01.30.Bb – Publications of lectures.PACS 03.67.Lx – Quantum computation.PACS 32.80.Qk – Coherent control of atomic interactions with photons.PACS 32.80.Lg – Mechanical effects of light on ions.PACS 32.80.Pj – Optical cooling of atoms; trapping.

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Contents

2 1. Introduction

3 2. Ion storage

3 2.1. Paul trap

4 2.2. Quantized motion

4 2.3. Choice of ion species

5 3. Laser interaction

7 3.1. Laser cooling

7 3.2. Initial state preparation

7 3.3. Quantum bits

10 3.4. Quantum gates

11 3.5. State detection

12 4. Experimental techniques

12 4.1. Laser pulses

12 4.2. Addressing individual ions in a string

13 4.3. State discrimination

14 4.4. Problems and solutions

15 5. Recent progress

17 6. New methods

17 6.1. Trap architecture

17 6.2. Sympathetic cooling

17 6.3. Fast gates

18 6.4. Qubits

18 7. Outlook: qubit interfacing

1. – Introduction

Quantum computing requires a composite quantum system that fulfils certain condi-tions: that it is well-isolated from environmental couplings, that its constituents can beindividually controlled and measured, and that there is a controllable interaction betweenthese constituents. These requirements have been concisely summarised by DiVincenzoin his set of criteria [1]. On the other hand, the experimental progress in the field ofquantum optics has spawned photonic and atomic systems which allow for quantum stateengineering, i.e. for on-demand preparation and manipulation of quantum properties ata very high degree of fidelity. Hence the application of quantum optical systems for theexperimental implementation of quantum information processing (QIP) resulted quitenaturally. Within this framework, trapped ions have been the most advanced systems,and they are among the most promising candidates for small- and medium scale quantumcomputation.

Single trapped ions were initially studied for their potential application for high-precision measurements and for the realisation of optical clocks, based on the possibility oflaser cooling [2] and on the idea of electron shelving [3]. In fact, the best optical oscillatorto date relies on a forbidden transition in a single cold mercury ion [4]. Parallel to thesedevelopments, single ions were identified as providing ideal conditions for demonstrating

Quantum computation with trapped ions 3

fundamental quantum optical phenomena, such as anti-bunching [5] and quantum jumps[6]. Due largely to progress in laser technology, a continuously increasing degree ofcontrol over the state of trapped ions has been achieved, such that what used to bespectroscopy has turned into coherent manipulation of the internal (electronic) state,while laser cooling has evolved into the control of the external degree of freedom, i.e. ofthe motional quantum state of the ion in the trap. Based on these developments, Ciracand Zoller proposed to use a trapped ion string for processing quantum information [7],and they described the operations required to realise a universal two-ion quantum logicalgate. This seminal proposal sparked intense experimental activities in several groups andhas led to spectacular results.

In these lecture notes, I will review the basic experimental techniques which enablequantum information processing with trapped ions. Without going into too many tech-nical details, I will explain how the fundamental concepts of quantum computing, suchas quantum bits (qubits), qubit rotations, and quantum gates translate into experimen-tal procedures in a quantum optics laboratory. Furthermore, the recent progress of iontrap quantum computing will be summarised, and some of the new approaches to meetfuture challenges will be mentioned, thus providing an update of the developments sincepublication of the previous lecture notes on this subject [8]. In most of what is presented,it is intended to provide an intuitive understanding of the matter that should enable thenon-specialist reader to appreciate the paradigmatic role and the potential of trappedions in the field of quantum computation. For more detailed explanations, referencesto the original work, to previous reviews, or to text books are provided. Examples andexperimental data are mostly taken from the work of the Innsbruck group [9], but pivotalwork is also carried out in other places [10].

2. – Ion storage

2.1. Paul trap. – An ionised atom in an electric field experiences very strong forces,corresponding to an acceleration on the order of 2 × 108 m/s2 per V/cm for a singlycharged calcium ion (Ca+). Although this cannot be directly employed to trap an ion ina static field, a rapidly oscillating electric field with quadrupole geometry,

Φ(r) = (U0 + V0 cos(ξt))x2 + y2 − 2z2

r20(1)

generates an effective (quasi-) potential which under suitably chosen experimental pa-rameters provides 3-dimensional trapping [11]. U0 and V0 are voltages. The field is madeto alternate so fast that the accelerated ion does not reach the electrodes of the trap be-fore it reverses its direction. In this case, the driven oscillatory motion at the frequencyξ of the applied field (micro-motion) has a kinetic energy which increases quadraticallywith the distance from the trap center, i.e. from the zero of the quadrupole field ampli-tude. By averaging over the micro-motion, an effective harmonic potential is obtained,in which the ion oscillates with a smaller frequency, the secular or macro-motion fre-quency ν; this is the principle of the Paul trap. The three frequencies νx,y,z along the

4 Jürgen Eschner

three principal axes of the trap may be different. The full trajectory of a trapped ion,consisting of the superimposed secular and micro-motion, is mathematically describedby a stable solution of a Mathieu-type differential equation [12]. Typical traps for singleions are roughly 1 mm in size, with a voltage V0 of several 100 V and frequency ξ of afew 10 MHz, leading to secular motion with frequencies ν in the low MHz range.

The linear variant of the Paul trap uses an oscillating field with linear quadrupole ge-ometry in two dimensions, providing dynamic confinement along those radial directions,while trapping in the third direction, i.e. along the axis of the 2-dimensional quadrupole,is obtained with a static field. This type of trap is advantageous for trapping several ionssimultaneously: under laser cooling and with radial confinement stronger than the axialone, the ions crystallise on the trap axis, thus minimising their micro-motion which oth-erwise leads to inefficient laser excitation and unwanted modulation-induced resonances(micro-motion sidebands). A schematic image of a linear Paul trap set-up is shown inFig. 1.

2.2. Quantized motion. – The secular motion of a trapped ion can be regarded as afree oscillation in a 3-dimensional harmonic trap potential, characterised by the threefrequencies νx,y,z. As will be described below, laser cooling reduces the thermal energyof that motion to values very close to zero. Therefore the motion must be accountedfor as a quantum mechanical degree of freedom. For a single ion it is described by theHamiltonians

Hk = h̄νk

(a†kak +

12

), (k = x, z, y)(2)

with energy eigenstates |nk〉 at the energy values h̄νk(nk + 1/2). The quantum regime ischaracterised by n being on the order of one or below.

For a string of N ions, there are 3N normal modes of vibration, 2N radial ones and Naxial ones. The two lowest-frequency axial modes are the center-of-mass mode, where allions oscillate like a rigid body, and the stretch mode, where each ion’s oscillation ampli-tude is proportional to its distance from the center [14, 15]. Their respective frequenciesare νz and

√3νz, independently of the number of ions, where νz is the axial frequency of

a single ion. Movies of ion strings with these modes excited can be found in [16]. As willbe explained below, coupling between different ions in a string for QIP purposes can beachieved by coherent laser excitation of transitions between the lowest quantum states|n = 0〉 and |n = 1〉 of one of the axial vibrational modes, usually the center-of-mass orthe stretch mode.

2.3. Choice of ion species. – The ion species which are suitable for QIP can be dividedinto two classes depending on the implementation of the qubits (see also section 3.3). Inall cases, a narrow optical transition connects two long-lived atomic states. One strategyis to use ions with a forbidden, direct optical transition, such as 40Ca+, 88Sr+, 138Ba+,172Yb+, 198Hg+, or 199Hg+. The other possibility is to employ the hyperfine structurein an odd isotope such as 9Be+, 25Mg+, 43Ca+, 87Sr+, 137Ba+, 111Cd+, 171Yb+, and

Quantum computation with trapped ions 5

Fig. 1. – Schematic illustration of a linear ion trap set-up with a trapped ion string. Thefour blades are on high voltage (neighbouring blades on opposite potential), oscillating at radiofrequency, thus providing Paul-type confinement in the radial directions. The tip electrodes areon positive high voltage and trap the ions axially. A laser addresses the ions individually andmanipulates their quantum state. The resonance fluorescence of the ions is imaged onto a CCDcamera. (Drawing courtesy of Rainer Blatt, Innsbruck.) In the lower part the CCD image of astring of eight cold, laser-excited ions is shown (from Ref. [13]). The distance between the outerions is about 70 µm.

to use an optical Raman transition between two hyperfine levels of the ground state.Both methods are currently pursued in different groups. For example, the NIST groupworks with 9Be+, the Michigan group with 111Cd+, and both the Innsbruck group andthe Oxford group use 40Ca+. Recent progress, such as Quantum Teleportation, has beenachieved with both types of qubits [17, 18].

3. – Laser interaction

Resonant interaction with laser light is used at all stages of QIP with trapped ions: forcooling, for initial state preparation, qubit manipulation, quantum gates, and for statedetection. Various types of optical transitions are employed, and the laser interacts withboth the electronic and motional quantum state. Photon scattering is either desired(for cooling and state detection) or unwanted (for qubit manipulations and quantum

6 Jürgen Eschner

Fig. 2. – Levels and transitions of 40Ca+. They are employed in the following ways for theprocesses described in this section: Doppler cooling happens by driving the S1/2 to P1/2 dipoletransition at 397 nm; a laser at 866 nm prevents optical pumping into the D3/2 level. Sidebandcooling is performed on the dipole-forbidden S1/2 to D5/2 line at 729 nm, with a laser at 854 nmpumping out the D5/2 level to enhance the cycling rate. A Zeeman sublevel of S1/2 and one ofD5/2 form the qubit (see right-hand diagram), on which coherent operations are driven by the729 nm laser. The final state is detected by the 397 nm laser which excites resonance fluorescencewhen the ion is in S1/2, but does not couple to D5/2.

gates). The relevant processes are summarised in this section, approximately in theorder in which they are applied in an experimental cycle: preparation, manipulation, i.e.logic operations, and measurement of the outcome. To give a specific example, Fig. 2shows the relevant levels, transitions, and wavelenghts of 40Ca+ and explains how theseprocesses are implemented in a real atom.

The atom-laser interaction processes can be understood by considering a simple two-level atom with resonance frequency ωA and states |g〉 (ground state) and |e〉 (excitedstate), and a single vibrational mode with frequency ν, states |n〉 and creation (annihi-lation) operator a† (a). The Hamiltonian of the interaction is

Hint = −h̄∆|e〉〈e|+ h̄νa†a + h̄Ω2

(|e〉〈g|eiη(a

†+a) + h.c.)

(3)

whereby ∆ = ωL−ωA is the detuning between laser frequency ωL and atomic resonanceand Ω is the Rabi frequency characterising the strength of the atom-laser coupling.The exponential in the bracket stands for the travelling wave of the laser, η(a† + a) =kz, with the Lamb-Dicke parameter η = kz0 which relates the spatial extension of themotional ground state z0 = 〈0|z2|0〉1/2 to the laser wavelength λ = 2π/k. Although notstrictly necessary, QIP experiments are typically performed in the Lamb-Dicke regime,characterised by η

√〈n〉 � 1, which means that the ion’s motional wave packet is much

smaller than the laser wavelength. Its consequence is that transitions |g, n〉 ↔ |e, n′〉

Quantum computation with trapped ions 7

between motional states of different quantum numbers n 6= n′ are suppressed as n −n′ increases, such that only changes n − n′ = 0,±1 need to be accounted for. Thecorresponding laser-driven transitions are called carrier (|g, n〉 → |e, n〉), red sideband(|g, n〉 → |e, n − 1〉), and blue sideband (|g, n〉 → |e, n + 1〉). With respect to Eq. (3)this corresponds to expanding the exponential, eiη(a

†+a) → 1 + iη(a† + a), which will beemployed in section 3.3.

3.1. Laser cooling . – Like the Paul trap mechanism, laser cooling is a pivotal ingredientfor preparing cold ions, in the Lamb-Dicke regime, for ion trap quantum computing. Lasercooling relies on the photon recoil or, more generally, the mechanical effect of light ina photon scattering process (absorption-emission cycle), which is caused by the spatialvariation of the electric field, described by the exponential in Eq. (3). Since severalcomprehensive reviews of laser cooling exist [19, 20, 21], I limit the description to a briefsummary of the most relevant techniques and their schematic explanation in Fig. 3.

Doppler cooling is the initial cooling stage. It is performed by exciting the ion(s) ona strong (usually a dipole) transition of natural linewidth Γ > ν, with the laser detunedby ∆ ' −Γ/2. It reduces the thermal energy to about h̄Γ in all vibrational modes whichhave an non-zero projection on the laser beam direction. With a typical dipole transitionof 20 MHz width and a trap frequency of 2 MHz, the average residual excitation of thevibrational quantum states is 〈n〉 ∼ 10. Much higher trap frequencies allow for Dopplercooling to 〈n〉 ∼ 1 [22].

Sideband cooling is the process which prepares a vibrational mode in the motionalground state |n = 0〉, which is a pure quantum state. It is accomplished by exciting anion on a narrow transition, Γ � ν, which in practical cases is either a forbidden opticalline, such as the S1/2 → D5/2 transition in Hg+ [23], Ca+ [24], or Sr+ [25], or a Ramantransition between hyperfine-split levels of the ground state, as in Be+ [22, 26] or Cd+

[27]. The laser is tuned into resonance with the red sideband transition at ∆ = −ν.To increase the cycling rate, the upper level |e〉 may be pumped out by driving anallowed transition to a higher-lying state which then decays back to |g〉. Pulsed schemesinvolving coherent Rabi flopping (π-pulses) are also used [22, 26]. Usually only one modeis cooled to the ground state, which then serves to establish coherent coupling betweenions in a string, see sections 3.3 and 3.4. Preparation of the motional ground state ofthe center-of-mass mode with > 99.9% probability has been achieved [24].

3.2. Initial state preparation. – With the motional state prepared, the atomic statemay have to be initialised, e.g. to a specific Zeeman sub-level of the ground state. Thisis achieved by applying a short optical pumping pulse of appropriate polarisation andbeam direction, e.g. in Ca+ by a σ+-polarised pulse at 397 nm which propagates alongthe quantisation axis and pumps all atomic population into S1/2,m = +1/2, see Fig. 2.

3.3. Quantum bits. – While state preparation relies on spontaneous emission, thequbit, in which the quantum information is encoded, requires two stable levels whichallow coherent laser excitation at Rabi frequencies much higher than all decay rates.Adequate states may be either a ground state and a metastable excited state connected

8 Jürgen Eschner

Fig. 3. – Laser cooling in the Lamb-Dicke regime, η � 1. Only δn = 0,±1 transitions arerelevant. (a) Energy levels and schematic representation of an absorption-emission cycle whichcools. (b) Spontaneous emission results in average heating because |n〉 → |n+1〉 transitions areslightly more probable than |n〉 → |n− 1〉 transitions. (c) In Doppler cooling, when Γ > ν, thelaser excites simultaneously |n〉 → |n〉 and |n〉 → |n± 1〉 transitions. By red-detuning the laserto ∆ ' −Γ/2, excitation from |g, n〉 to |e, n − 1〉 is favoured which thus provides cooling. Thefinal state of Doppler cooling is an equilibrium between these cooling and heating processes.(d) In sideband cooling, when Γ � ν, the laser is tuned to the |g, n〉 to |e, n − 1〉 sidebandtransition at ∆ = −ν, such that all other transitions are off-resonant. This provides a muchstronger cooling effect than Doppler cooling, while the heating by spontaneous emission is thesame. Thus the final state is very close to the motional ground state.

by a forbidden optical transition, as in 40Ca+ or 88Sr+ (”optical qubit”), or two hyperfinesub-levels of the ground state of an ion with non-zero nuclear spin, as in 9Be+, 111Cd+,or 171Yb+ (”hyperfine qubit”). Depending on the magnetic properties of the ion species,particular Zeeman substates of the levels are selected, like S1/2,m = 1/2 and D5/2,m =5/2 in Ca+ (Fig. 2). An optical qubit transition is driven with a single strong laser withΩ � Γ, while in a hyperfine qubit two lasers, far detuned from an intermediate level,drive a Raman transition with Ω being the Raman Rabi frequency [28]. Both cases canagain be treated as an atomic two-level system {|g〉, |e〉}.

These atomic qubit levels are combined with the motional qubit, consisting of the twolowest levels |0〉 and |1〉 of the vibrational mode which has been cooled to the ground state.Considering first a single trapped ion, then the basis for quantum logical operations, the”computational subspace” (CS), is formed by the four states {|g, 0〉, |g, 1〉, |e, 0〉, |e, 1〉}.

Quantum computation with trapped ions 9

Fig. 4. – Computational subspace of a single ion with carrier (C) and sideband (±) transitions,and example for the evolution of a quantum state under application of π/2-pulses. |S〉 and |D〉stand for |g〉 and |e〉 in the text, indicating the implementation with Ca+. Note the sign changeof the wave function after a 2π-pulse.

For coherent manipulation of the quantum state within the CS, the red sideband, carrier,or blue sideband transition may be excited by tuning the laser to ∆ = −ν, 0, or +ν,respectively.

The interaction Hamiltonians for these three excitation processes are derived fromEq. (3) in the Lamb-Dicke regime and after neglecting non-resonant terms,

(red sideband) H− = h̄ηΩ2

(|e〉〈g|a + h.c.)(4)

(carrier) HC = h̄Ω2

(|e〉〈g|+ h.c.)(5)

(blue sideband) H+ = h̄ηΩ2

(|e〉〈g|a† + h.c.

).(6)

These Hamiltonians induce unitary dynamics in the CS, which are described by theoperators

(red sideband) R−(θ, φ) = exp[iθ

2(eiφ|e〉〈g|a + e−iφ|g〉〈e|a†

)](7)

(carrier) RC(θ, φ) = exp[iθ

2(eiφ|e〉〈g|+ e−iφ|g〉〈e|

)](8)

(blue sideband) R+(θ, φ) = exp[iθ

2(eiφ|e〉〈g|a† + e−iφ|g〉〈e|a

)],(9)

where θ = Ωt (θ = ηΩt on the sidebands) is the rotation angle of a pulse of duration t,and φ is the phase of the laser. The laser phase is arbitrary when the first of a seriesof pulses is applied on one transition, but it has to be kept track of in all subsequentoperations until the final state measurement. Fig. 4 shows an example how the quantumstate evolves in the CS under the action of subsequent R+(π2 , 0) pulses.

An example for coherent state manipulation, or unitary qubit rotations, within theCS of a single ion is shown in Fig. 5. This result was obtained with an optical qubit in asingle 40Ca+ ion. Analogous operations with a hyperfine qubit in 9Be+ were employed torealise the first single-ion quantum gate between an internal and a motional qubit [29].

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Fig. 5. – Coherent time evolution of the quantum state in the CS of a single ion. After preparationof a single Ca+ ion in the ground state |g, 0〉, a laser pulse of certain duration is applied, tuned tothe carrier (top) or blue sideband (bottom) transition. At the end of the pulse, the populationof the excited state is measured (see section 3

.5). Points are measured data, the line is a fit.

Note the different time scales of the two plots; the Rabi frequency on the sideband is smaller bya factor of η. From Ref. [30].

3.4. Quantum gates. – The central concept around which quantum gates with trappedions are constructed are the common modes of vibration. Sideband excitation of one ionmodifies the state of the whole string and has an effect on subsequent sideband interactionwith all other ions, thus providing the required ion-ion coupling. Due to the strongCoulomb coupling between the ions, the vibrational modes are non-degenerate, thereforethey can be spectrally addressed, by tuning the laser to the sideband of one particularmode. In particular, the center-of-mass mode at the lowest frequency ν is separatedfrom the next adjacent mode, the stretch mode, by (

√3− 1)ν. Example spectra can be

found in Refs. [31, 32]. Spectral resolution of the sidebands imposes a speed limit ongate operations, which will be discussed in section 6.3.

The generalisation of the computational subspace from the single-ion case is straight-forward: the CS of two ions (these can be any two in a longer string) is the productspace of their internal qubits and the selected vibrational mode. The resulting statesand transitions are displayed in Fig. 6.

The role of the vibrational mode is illustrated by considering a basic entangling oper-ation which can be realised with two laser pulses that address the ions individually (seealso section 4.2). The pulses are defined analogously to Eqs. (7-9), with the additional

Quantum computation with trapped ions 11

Fig. 6. – Computational subspace of two ions and one vibrational mode, with all possible transi-tions. Short, medium, and long arrows are red sideband, carrier, and blue sideband transitions,respectively. As before, |S〉 and |D〉 stand for |g〉 and |e〉 in the text and refer to the exampleof a Ca+ ion.

superscript (1) or (2) indicating the addressed ion,

|g, g, 0〉 R(1)+ (

π2 , 0)

−→1√2

(|g, g, 0〉+ i|e, g, 1〉) R(2)− (π, 0)−→

1√2

(|g, g, 0〉 − |e, e, 0〉) .

The pulse sequence creates an entangled state (|g, g〉 − |e, e〉)/√

2 of the two internalqubits while leaving the motional qubit in the ground state.

Several specific gate protocols within the two-ion CS have been proposed and im-plemented. The classic Cirac-Zoller gate works in the following way [7]: first the stateof ion 1 is transferred into the motional qubit by a SWAP operation (see section 4.4).Then a single-ion gate is carried out between the motional qubit and ion 2. Finally themotional qubit state is swapped back into ion 1, which completes the gate between thetwo ions. The experimental realisations of quantum gates are discussed in section 5.

Note that the motion, although it could be regarded as providing one qubit pervibrational mode, always serves only as a ”bus” connecting the internal qubits of theions. In particular, the motional qubits cannot be measured independently. Thus at theend of a logic operation, the motional state has to factorise from the state of the internalqubits, which itself can be entangled.

3.5. State detection. – At the end of a quantum logical process the state of the qubitshas to be determined. This is accomplished by exciting the ions and recording the scat-tered fluorescence photons on a transition which couples to only one of the qubit levels;see Fig. 2 for the example of Ca+. More technical details are given in the next section.State detection is a projective measurement which destroys all quantum correlations.The outcome of one individual measurement is a binary signal for each ion, full fluores-cence or no photons, corresponding to either |g〉 or |e〉. For finding out the probabilitiesthat the ion is in either of the two states after the logic operation, one has to repeat the

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Fig. 7. – Creation of laser pulses. Continuous light from a stable laser (coherence time � gatetime) is sent through an acousto-optical modulator (AOM), which receives a radio frequencysignal from a stable oscillator (RF). The RF frequency ∆ is added to the light frequency, thusdefining the laser detuning, the RF phase φ translates directly into the phase of the light, andthe RF power P controls the light intensity I, including switching it on and off. The AOMprocess is based on Bragg diffraction from a sound wave in a crystal.

measurement many times. Typically 100 or several 100 runs are used. In the example ofFig. 5, each data point is an average of 200 single measurements.

4. – Experimental techniques

4.1. Laser pulses. – The experimental conditions for qubit manipulations are stringent.Even for a basic operation like a two-ion quantum gate, a sequence of several pulses isrequired, during which phase-perturbing or decay processes must be negligible. Apartfrom stable atomic states, this requires the control of environmental magnetic fields, andin particular it puts challenging requirements on the stability of the laser sources. In theexample of the optical qubit, where spontaneous decay of the excited state happens onthe 1 s time scale, the laser phase fluctuations are critical. A residual laser linewidthin the 100 Hz range or better is desired in order to maintain phase coherence over asignificant number of pulses. To achieve this, sophisticated stabilisation techniques likethose developed in the context of optical clocks have to be applied.

With hyperfine qubits, laser stability is still very important but less critical, becausethere the difference frequency between two light fields is relevant, which is in the GHzrange and can be made very stable using microwave oscillators.

Once the phase-stable light field for driving the qubits is generated, the creation oflaser pulses of desired amplitude, detuning, phase shift, and duration is comparativelysimple. It is illustrated in Fig. 7.

4.2. Addressing individual ions in a string. – The conventional way to manipulateindividual qubits in a string of ions, as envisioned in the original Cirac-Zoller proposal, is

Quantum computation with trapped ions 13

Fig. 8. – Excitation probability on the qubit transition measured on a three-ion string whenthe laser beam is scanned across it (unpublished data from Innsbruck). Both axes are linearscales. The data points are fitted by the sum of three Gaussian functions. From the individualGaussians of about 2.5 µm width (shown as thin lines) one finds that the intensity on an adjacention is ∼ 1/400 that on the addressed ion.

to use a well-focused laser beam which interacts with only one ion at a time, and whichcan be directed at any ion in the string. This kind of addressing was first demonstratedin [33] and has been employed in the first 2-ion gate implementation with optical qubits[34]. It is only possible if the distance between the ions is significantly larger than thediffraction limit of the focussing lens. This limits the range of possible trap frequenciesand thus the speed of gate operations [35]. With the 729 nm light exciting the opticalqubit in Ca+, focusing to ∼ 2.5 µm can typically be achieved, leading to an ion spacingaround 5 µm. Fig. 8 shows an example. The Rabi frequency on a neighbouring ioncan reach about 5% of that on the addressed ion, introducing addressing errors in thequantum logic operations. Such systematic errors could, however, be compensated forby adjustments to pulse lengths and phases [36].

Addressing in gate operations is not indispensable, since protocols have been devel-oped [37] and demonstrated [38, 39] which employ simultaneous laser interaction withboth participating ions. The gate used by the NIST group in their recent experimentsrelies on the action of spin-dependent dipole forces on both ions when they are simul-taneously excited [40]. Still, in general a process is required which allows for shieldingone ion from the light which excites the adjacent one. The strategy which appears to bemost promising is to separate and combine ions in segmented traps, see section. 6.1.

4.3. State discrimination. – The final state read-out on a string of ions, by a measure-ment as described in section 3.5, requires recording the resonance fluorescence of eachion in the string. At the same time, state read-out should happen fast to allow for ahigh experimental cycle frequency and to avoid state changes by decay processes. Thushigh photon detection efficiency at low dark count rate is desired. A photon counting

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Fig. 9. – Example histogram for state discrimination with a single Ca+ ion (c.f. [41]). The ion isprepared with ∼ 70% probability in the S1/2 state, which fluoresces, and with ∼ 30% probabilityin D5/2, which is dark. Resonance fluorescence photons are collected for 9 ms. The experimentis repeated several 1000 times. The separation between the two Poisson distributions allows todefine a clear threshold that discriminates between the possible outcomes.

camera resolving the individual ions is the appropriate device. The suitable parametersare found by acquiring a histogram of photon counts corresponding to the positions ofthe ions on the camera image and to the two possible signal levels. The histogram willshow Poisson distributions around the mean numbers of fluorescence and dark counts,and efficient state discrimination is achieved when the distributions do not overlap. Asshown in the example of Fig. 9, a few 10 photons are sufficient. In the example of Ca+,in 10 ms a state discrimination efficiency above 99% can be reached [41].

In experiments without individual addressing, the ions have to be separated usingsegmented traps for measuring their final states individually [18].

4.4. Problems and solutions. – Some additional experimental tricks are mentioned inthis section, in order to give examples for experimental problems and their solutions.Further important technological progress is discussed in section 6.1.

Composite pulses. The computational subspace defined in section 3.3 is not closed:sideband pulses couple to states outside the CS, namely |g, 1〉 to |e, 2〉 on the blue and|e, 1〉 to |g, 2〉 on the red sideband. Moreover the Rabi frequency on the |1〉 ↔ |2〉transitions is larger than on |0〉 ↔ |1〉 by a factor of

√2. This problem can be solved

by composite pulses. This technique is well-known in NMR spectroscopy [42] and itsapplication for ion traps has been proposed in Ref. [43]. An application is the compositeSWAP gate. The SWAP exchanges the state of two qubits and is employed, e.g., to write

Quantum computation with trapped ions 15

the state of the internal qubit into the motional qubit, like in

(a|g〉+ b|e〉)(c|1〉+ d|0〉) → (c|g〉+ d|e〉)(a|1〉+ b|0〉) .

It can be obtained by a π-pulse on the blue sideband, but only if the initial state is not|g, 1〉. A composite pulse sequence,

RSWAP = R+(π/√

2, 0) R+(π√

2, φSWAP ) R+(π/√

2, 0) ,

where φSWAP = cos−1(cot2(π/√

2)) ≈ 0.303π ,

performs the SWAP operation for all states of the CS, by acting as a π-pulse on the|g, 0〉 ↔ |e, 1〉 transition and simultaneously as a 4π rotation on |g, 1〉 ↔ |e, 2〉. Compositepulses have been used in the implementation of the Cirac-Zoller quantum gate with Ca+

ions [34].Qubit hiding. In a teleportation experiment, where part of the operations is condi-

tional on the outcome of an intermediate measurement, it is necessary to measure thestate of one ion while the other ions are in superposition states. Therefore these ionshave to be protected from the resonant light. This is achieved either by separating theions spatially, which has been shown to be possible without affecting internal-qubit su-perpositions [44, 18], or by a hiding pulse, as has been demonstrated with Ca+ [17, 45].Such a π-pulse transfers the population from the qubit level in S1/2, which might coupleto the resonant light on the neighbouring ion, to a Zeeman sublevel of D5/2 which is dif-ferent from the qubit level. After the measurement pulse on the adjacent ion, a −π-pulsereverses the hiding process and restores the superposition of the qubit.

Further examples for techniques which deal with experimental imperfections are spinecho pulses to counteract the effect of magnetic field fluctuations and gradients [18],and the compensation of Stark shifts which occur during sideband excitation due tonon-resonant interaction with the carrier transition [46].

5. – Recent progress

After preliminary work which already implemented the relevant entangling operations[38, 39], the first two-ion gates were realised in 2003, simultaneously in Boulder with ahyperfine [40] and in Innsbruck with an optical qubit [34]. The optical implementationwith Ca+ ions was very close to the original Cirac-Zoller proposal, while the hyperfineversion with Be+ used a protocol based on state-dependent forces, similar to the schemeproposed by Sørensen and Mølmer [37]. Figure 10 displays the result of the Innsbruckexperiment.

A spectacular result in 2004 was the first deterministic quantum teleportation of thestate of an atom, which again was achieved by the same two groups simultaneously[18, 17]. A diagram which illustrates the procedure nicely can be found in [47]. TheNIST experiment involved shuttling ions between zones of a segmented trap, in order to

16 Jürgen Eschner

Fig. 10. – Experimental implementation of the Cirac-Zoller two-ion quantum CNOT gate withCa+, from [34]. Each pair of plots shows for both ions the time evolution of the populationsof the qubit levels. The initial states are the four basis states, |SS〉, |SD〉, |DS〉, and |DD〉.With initial state |S〉 in ion 1, the state of ion 2 remains the same (left side), whereas initialstate |D〉 in ion 1 causes ion 2 to change its state (right side). Each circle is an average of100 measurements, where the gate pulse sequence was interrupted at that time and a statemeasurement was applied to both ions. Each shaded area is a laser pulse or pulse sequence (seetop right), starting with the preparation of both ions, then the SWAP operation between ion 1and the motional qubit; the central pulse sequence on ion 2 is the CNOT gate between its stateand the motion, using composite pulses; then the SWAP is reversed on ion 1, which completesthe gate.

allow for both individual and simultaneous laser manipulation of the ions. This can beconsidered as a first step towards scalable trap architecture.

The first realisation of a decoherence-free subspace was reported in Ref. [39]. Morerecently, very long coherence times, of many seconds, of entangled states and states indecoherence-free subspaces have been demonstrated in Refs. [48] and [49].

Several basic quantum logical operations were also demonstrated, such as quantumdense coding [50], quantum error correction [51], and the quantum Fourier transform[52]. Specific entangled 2-ion [53] and 3-ion [54] entangled quantum states have beendeterministically created and characterised by quantum tomography. The latest mile-stone result is the creation and characterisation of multi-ion entangled states (W states)

Quantum computation with trapped ions 17

including an entangled ”quantum byte” formed by 8 ions [55].

6. – New methods

After basic few-ion operations have been demonstrated and simple quantum algo-rithms have been implemented, the question of scalability has to be addressed seriously.One very important aspect of it is error correction, which is discussed in great detailin A. Steane’s contribution to this volume [56]. The scheme he presents of an ion trapquantum computer which implements full error correction appears to be an overwhelmingtask. In this final section I summarise a few recent ideas and results which are expectedto contribute to meeting this challenge. Three main directions are considered, ion shut-tling in segmented traps, sympathetic cooling, and fast gates. Furthermore, I discusssome issues concerning the choice of the optimal qubit.

6.1. Trap architecture. – Addressing ions in a static string by steering a laser beambecomes technically difficult as the number of ions increases. Moreover, the conven-tional two-ion gate protocol employing a vibrational mode of a string becomes difficultto implement as the size of the string grows, because the increasing mass of the ion crys-tal (entering into the Lamb-Dicke parameter η) reduces the coupling on the sidebandtransitions. Therefore a trap architecture seems favourable which allows for moving theions. In this scenario, ions are picked from a register and shuttled by time-dependenttrap voltages into an interaction zone where two-ion gates are carried out of the kindused in [18]. A detailed scheme has been proposed by the NIST group [57], and severalother groups have started activities to simulate, build, and test various implementations[58, 59, 60, 61, 9, 10]. A notable experimental achievement has very recently been re-ported by the Michigan group who shuttled an ion around a corner in a tee-junction [62].

6.2. Sympathetic cooling . – Part of the future trap technology will be to apply newmeans of cooling the ions without perturbing their quantum state, i.e. without scatteringphotons from them. This can be achieved by sympathetic cooling, when the collectivemodes of a string are cooled by scattering photons only from ions which do not participatein the quantum logic process. Sympathetic Doppler cooling was demonstrated some timeago with Hg+-Be+ mixtures [64] and with Mg+ [65]. Ground state cooling of a two-ioncrystal through one of the ions was reported in [32]. Further relevant results were obtainedwith mixtures of Cd+ isotopes [63] and with a Be+-Mg+ crystal which was cooled tothe motional ground state [66]. Very recently, sympathetic cooling and quantum logichave been combined in an experiment with a Be+-Al+ ion crystal, where high-precisionspectroscopy was carried out on the Al+ ion, while cooling and state read-out were doneon the Be+ ion [67].

6.3. Fast gates. – Another major contribution to scaling up ion trap QIP is expectedfrom faster gate operations. The current limitation is the use of spectrally resolvedsideband resonances, which requires individual laser pulses to be longer than the inversetrap frequency. Smaller traps with larger frequencies can improve the situation, but

18 Jürgen Eschner

theoretical and numerical studies indicate that one could in fact use significantly shorter,suitably designed pulses or pulse trains [68, 69, 70]. Their action is, rather than toexcite one mode, that they impart photon momentum kicks and thus interact with asuperposition of many modes or even with the local modes of one ion, i.e. its localvibration as if all other ions were fixed [70]. It will be interesting to see how ideas ofquantum coherent control can help advancing into this direction [71].

6.4. Qubits. – Many issues have to be considered to identify the optimum type of qubitand the optimum ion species. Using an optical qubit in an ion without hyperfine structuremakes the qubit levels sensitive to ambient magnetic fields and to laser phase fluctuations,which are sources of decoherence. Here, effective qubits can be constructed from multi-ionquantum states, which form a decoherence-free subspace [49]. With hyperfine qubits theeffect of laser phase noise is reduced because the two fields for the Raman transition arederived from the same source, and their difference frequency is stabilised to a microwaveoscillator. The magnetic field sensitivity can also be significantly decreased by usingm = 0 magnetic sublevels [72], or by applying a static magnetic field at which thequbit levels have the same differential Zeeman shift [48]. The limitation by spontaneousscattering during a Raman excitation pulse has been assessed in [28], pointing at heavyions like Cd+ or Hg+ as good candidates. From the practical point of view, the hyperfinequbit in 43Ca+ is an attractive choice for the accessibility of the Ca+ transitions bylaser sources. Work with 43Ca+ has been started in Innsbruck and Oxford [73]. Directexcitation of hyperfine qubits by microwave radiation has also been considered [74].It requires the presence of high magnetic field gradients to obtain sufficient sidebandcoupling (through an effective Lamb-Dicke parameter), which at the same time wouldsplit the frequencies of neighbouring ions, thus allowing for their addressing.

7. – Outlook: qubit interfacing

An important field of activity related to QIP with trapped ions is qubit interfacing,i.e. linking static quantum information stored in ionic qubits to photonic quantum states.As an outlook for the topics discussed so far, I will discuss that subject briefly in thisfinal section.

The quantum interaction between single trapped ions and single photons is a centralobjective for many schemes connected to QIP with ions, such as quantum networking[75], distributed quantum computing, or quantum communication. Coherent coupling ofionic qubits to photonic modes may be achieved in a high-finesse resonator. Entangledion-photon states and entangled states of distant ions can also be generated by projectivestate preparation conditioned on photon detection events. I review briefly the existingapproaches and results with ion traps. Important and highly advanced work in this fieldis also carried out with trapped neutral atoms [76].

The direct coupling of a trapped ion to a high-finesse optical cavity has been realisedin two experiments. The MPQ group demonstrated coupling on a dipole transition[77, 78], which was then applied to create single photons with tailored wave packets [79].

Quantum computation with trapped ions 19

Fig. 11. – Schematic experimental set-up for postselective entanglement creation between distanttrapped ions (after [83]). After preparation of both ions in a state |0〉, a laser pulse excites astate-changing transition |0〉 → |1〉 with small probability �, thus producing the state |0, 0〉 +�(|0, 1〉+ |1, 0〉) + �2|1, 1〉. Part of the scattered light from both ions is coherently superimposedon a beam splitter. A photon detection event behind the beam splitter (of π polarisation in theexample) does not allow to determine which ion changed its state, and therefore signals creationof the entangled state (|0, 1〉+ |1, 0〉)/

√2.

In Innsbruck, Rabi oscillations on a qubit transition induced by a coherent cavity fieldwere observed, and cavity sideband excitation was demonstrated [80, 81].

The entanglement between the polarisation of an ion and an emitted photon has beendemonstrated in an experiment of the Michigan group [82]. Based on this result, andfollowing proposals by Cabrillo and others [83], several experiments pursue the postse-lective creation of entangled states of distant ions. It relies on the indistinguishability ofphoton scattering from the two ions, as explained in Fig. 11.

The possible combination of local quantum logic operations in ion traps with ion-photon interfaces which establish connections between distant traps is certainly an ex-citing additional aspect of the future perspectives of ion trap quantum computing.

20 Jürgen Eschner

∗ ∗ ∗

I would like to thank the directors and organisers of the International School ofPhysics ”Enrico Fermi”, G. Casati, D. L. Shepelyansky, P. Zoller, and G. Benenti forgiving me the opportunity to present these lectures. I thank Christian Roos for veryhelpful comments, Rainer Blatt for allowing me to use his drawing of a linear ion trap,and the Innsbruck team for the results which I used in my examples. The work inInnsbruck is supported by the Austrian Science Fund (FWF, special research centerSFB 15), by the European Commission (CONQUEST and QGATES networks, MRTN-CT-2003-505089 and IST-2001-38875), by the ARO (DAAD19-03-1-0176), and by theInstitut für Quanteninformation GmbH.

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