Quantum Computation with Trapped Ions

Quantum computation with trapped ions

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

Summary. — The development, status, and challenges of ion trap quantum computing are reviewed at a tutorial level for non-specialists. Particular emphasis is put on explaining the experimental implementation of the most fundamental concepts and 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. 2 Jurgen¨ Eschner

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 conditions: that it is well-isolated from environmental couplings, that its constituents can be individually controlled and measured, and that there is a controllable interaction between these constituents. These requirements have been concisely summarised by DiVincenzo in his set of criteria [1]. On the other hand, the experimental progress in the field of quantum optics has spawned photonic and atomic systems which allow for quantum state engineering, i.e. for on-demand preparation and manipulation of quantum properties at a very high degree of fidelity. Hence the application of quantum optical systems for the experimental implementation of quantum information processing (QIP) resulted quite naturally. Within this framework, trapped ions have been the most advanced systems, and they are among the most promising candidates for small- and medium scale quantum computation. 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 of laser cooling [2] and on the idea of electron shelving [3]. In fact, the best optical oscillator to date relies on a forbidden transition in a single cold mercury ion [4]. Parallel to these developments, 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 of control over the state of trapped ions has been achieved, such that what used to be spectroscopy 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. of the motional quantum state of the ion in the trap. Based on these developments, Cirac and 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 logical gate. This seminal proposal sparked intense experimental activities in several groups and has led to spectacular results. In these lecture notes, I will review the basic experimental techniques which enable quantum information processing with trapped ions. Without going into too many technical details, I will explain 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 ion trap quantum computing will be summarised, and some of the new approaches to meet future challenges will be mentioned, thus providing an update of the developments since publication 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 the non-specialist reader to appreciate the paradigmatic role and the potential of trapped ions in the field of quantum computation. For more detailed explanations, references to the original work, to previous reviews, or to text books are provided. Examples and experimental data are mostly taken from the work of the Innsbruck group [9], but pivotal work 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 singly charged calcium ion (Ca+). Although this cannot be directly employed to trap an ion in a static field, a rapidly oscillating electric field with quadrupole geometry,

x2 + y2 − 2z2 (1) Φ(r) = (U0 + V0 cos(ξt)) 2 r0 generates an effective (quasi-) potential which under suitably chosen experimental parameters provides 3-dimensional trapping [11]. U0 and V0 are voltages. The field is made to alternate so fast that the accelerated ion does not reach the electrodes of the trap before it reverses its direction. In this case, the driven oscillatory motion at the frequency ξ of the applied field (micro-motion) has a kinetic energy which increases quadratically with the distance from the trap center, i.e. from the zero of the quadrupole field amplitude. 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 frequency ν; this is the principle of the Paul trap. The three frequencies νx,y,z along the 4 Jurgen¨ 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 described by a stable solution of a Mathieu-type differential equation [12]. Typical traps for single ions are roughly 1 mm in size, with a voltage V0 of several 100 V and frequency ξ of a few 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 geometry 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 ions simultaneously: under laser cooling and with radial confinement stronger than the axial one, 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 in Fig. 1. . 2 2. Quantized motion. – The secular motion of a trapped ion can be regarded as a free oscillation in a 3-dimensional harmonic trap potential, characterised by the three frequencies νx,y,z. As will be described below, laser cooling reduces the thermal energy of that motion to values very close to zero. Therefore the motion must be accounted for as a quantum mechanical degree of freedom. For a single ion it is described by the Hamiltonians

1 H = ¯hν a† a + , (k = x, z, y)(2) k k k k 2 with energy eigenstates |nki at the energy values ¯hνk(nk + 1/2). The quantum regime is characterised 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 N axial ones. The two lowest-frequency axial modes are the center-of-mass mode, where all ions oscillate like a rigid body, and the stretch mode, where each ion’s oscillation amplitude is proportional√ to its distance from the center [14, 15]. Their respective frequencies are 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 will be explained below, coupling between diﬀerent ions in a string for QIP purposes can be achieved by coherent laser excitation of transitions between the lowest quantum states |n = 0i and |n = 1i of one of the axial vibrational modes, usually the center-of-mass or the stretch mode. . 2 3. Choice of ion species. – The ion species which are suitable for QIP can be divided . into two classes depending on the implementation of the qubits (see also section 3 3). In all cases, a narrow optical transition connects two long-lived atomic states. One strategy is 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 structure in 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. The four blades are on high voltage (neighbouring blades on opposite potential), oscillating at radio frequency, thus providing Paul-type conﬁnement in the radial directions. The tip electrodes are on positive high voltage and trap the ions axially. A laser addresses the ions individually and manipulates their quantum state. The resonance ﬂuorescence of the ions is imaged onto a CCD camera. (Drawing courtesy of Rainer Blatt, Innsbruck.) In the lower part the CCD image of a string of eight cold, laser-excited ions is shown (from Ref. [13]). The distance between the outer ions 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 group works with 9Be+, the Michigan group with 111Cd+, and both the Innsbruck group and the Oxford group use 40Ca+. Recent progress, such as Quantum Teleportation, has been achieved 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: for cooling, for initial state preparation, qubit manipulation, quantum gates, and for state detection. Various types of optical transitions are employed, and the laser interacts with both the electronic and motional quantum state. Photon scattering is either desired (for cooling and state detection) or unwanted (for qubit manipulations and quantum 6 Jurgen¨ Eschner

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

gates). The relevant processes are summarised in this section, approximately in the order 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. 2 shows the relevant levels, transitions, and wavelenghts of 40Ca+ and explains how these processes 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 |gi (ground state) and |ei (excited state), and a single vibrational mode with frequency ν, states |ni and creation (annihi- lation) operator a† (a). The Hamiltonian of the interaction is

Ω † (3) H = −¯h∆|eihe| +hνa ¯ †a +h ¯ |eihg|eiη(a +a) + h.c. int 2 whereby ∆ = ωL − ωA is the detuning between laser frequency ωL and atomic resonance and 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 the 2 1/2 motional ground state z0 = h0|z |0i to the laser wavelength λ = 2π/k. Although not strictly necessary, QIP experiments are typically performed in the Lamb-Dicke regime, characterised by ηphni 1, which means that the ion’s motional wave packet is much smaller than the laser wavelength. Its consequence is that transitions |g, ni ↔ |e, n0i Quantum computation with trapped ions 7 between motional states of different quantum numbers n 6= n0 are suppressed as n − n0 increases, such that only changes n − n0 = 0, ±1 need to be accounted for. The corresponding laser-driven transitions are called carrier (|g, ni → |e, ni), red sideband (|g, ni → |e, n − 1i), and blue sideband (|g, ni → |e, n + 1i). With respect to Eq. (3) † this corresponds to expanding the exponential, eiη(a +a) → 1 + iη(a† + a), which will be . employed in section 3 3. . 3 1. Laser cooling. – Like the Paul trap mechanism, laser cooling is a pivotal ingredient for preparing cold ions, in the Lamb-Dicke regime, for ion trap quantum computing. Laser cooling relies on the photon recoil or, more generally, the mechanical effect of light in a photon scattering process (absorption-emission cycle), which is caused by the spatial variation of the electric field, described by the exponential in Eq. (3). Since several comprehensive reviews of laser cooling exist [19, 20, 21], I limit the description to a brief summary 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) on a strong (usually a dipole) transition of natural linewidth Γ > ν, with the laser detuned by ∆ ' −Γ/2. It reduces the thermal energy to abouth ¯Γ in all vibrational modes which have an non-zero projection on the laser beam direction. With a typical dipole transition of 20 MHz width and a trap frequency of 2 MHz, the average residual excitation of the vibrational quantum states is hni ∼ 10. Much higher trap frequencies allow for Doppler cooling to hni ∼ 1 [22]. Sideband cooling is the process which prepares a vibrational mode in the motional ground state |n = 0i, which is a pure quantum state. It is accomplished by exciting an ion on a narrow transition, Γ ν, which in practical cases is either a forbidden optical + + + line, such as the S1/2 → D5/2 transition in Hg [23], Ca [24], or Sr [25], or a Raman transition 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 |ei may be pumped out by driving an allowed transition to a higher-lying state which then decays back to |gi. Pulsed schemes involving coherent Rabi flopping (π-pulses) are also used [22, 26]. Usually only one mode is cooled to the ground state, which then serves to establish coherent coupling between . . ions in a string, see sections 3 3 and 3 4. Preparation of the motional ground state of the center-of-mass mode with > 99.9% probability has been achieved [24]. . 3 2. Initial state preparation. – With the motional state prepared, the atomic state may have to be initialised, e.g. to a specific Zeeman sub-level of the ground state. This is achieved by applying a short optical pumping pulse of appropriate polarisation and beam direction, e.g. in Ca+ by a σ+-polarised pulse at 397 nm which propagates along the 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, the qubit, in which the quantum information is encoded, requires two stable levels which allow 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 Jurgen¨ Eschner

Fig. 3. – Laser cooling in the Lamb-Dicke regime, η 1. Only δn = 0, ±1 transitions are relevant. (a) Energy levels and schematic representation of an absorption-emission cycle which cools. (b) Spontaneous emission results in average heating because |ni → |n + 1i transitions are slightly more probable than |ni → |n − 1i transitions. (c) In Doppler cooling, when Γ > ν, the laser excites simultaneously |ni → |ni and |ni → |n ± 1i transitions. By red-detuning the laser to ∆ ' −Γ/2, excitation from |g, ni to |e, n − 1i is favoured which thus provides cooling. The final 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, ni to |e, n − 1i sideband transition at ∆ = −ν, such that all other transitions are off-resonant. This provides a much stronger cooling effect than Doppler cooling, while the heating by spontaneous emission is the same. 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 hyperfine sub-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 with Ω 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 can again be treated as an atomic two-level system {|gi, |ei}. These atomic qubit levels are combined with the motional qubit, consisting of the two lowest levels |0i and |1i 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, 0i, |g, 1i, |e, 0i, |e, 1i}. 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. |Si and |Di stand for |gi and |ei in the text, indicating the implementation with Ca+. Note the sign change of 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 from Eq. (3) in the Lamb-Dicke regime and after neglecting non-resonant terms,

θ (7) (red sideband) R (θ, φ) = exp i eiφ|eihg|a + e−iφ|gihe|a† − 2 θ (8) (carrier) R (θ, φ) = exp i eiφ|eihg| + e−iφ|gihe| C 2 θ (9) (blue sideband) R (θ, φ) = exp i eiφ|eihg|a† + e−iφ|gihe|a , + 2 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 series of pulses is applied on one transition, but it has to be kept track of in all subsequent operations until the final state measurement. Fig. 4 shows an example how the quantum π state evolves in the CS under the action of subsequent R+( 2 , 0) pulses. An example for coherent state manipulation, or unitary qubit rotations, within the CS of a single ion is shown in Fig. 5. This result was obtained with an optical qubit in a single 40Ca+ ion. Analogous operations with a hyperfine qubit in 9Be+ were employed to realise the first single-ion quantum gate between an internal and a motional qubit [29]. 10 Jurgen¨ Eschner

Fig. 5. – Coherent time evolution of the quantum state in the CS of a single ion. After preparation of a single Ca+ ion in the ground state |g, 0i, a laser pulse of certain duration is applied, tuned to the carrier (top) or blue sideband (bottom) transition. At the end of the pulse, the population . of 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 by a factor of η. From Ref. [30].

. 3 4. Quantum gates. – The central concept around which quantum gates with trapped ions are constructed are the common modes of vibration. Sideband excitation of one ion modifies the state of the whole string and has an effect on subsequent sideband interaction with all other ions, thus providing the required ion-ion coupling. Due to the strong Coulomb coupling between the ions, the vibrational modes are non-degenerate, therefore they can be spectrally addressed, by tuning the laser to the sideband of one particular mode. In particular, the center-of-mass mode at the√ lowest frequency ν is separated from 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 on . gate 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 product space of their internal qubits and the selected vibrational mode. The resulting states and transitions are displayed in Fig. 6. The role of the vibrational mode is illustrated by considering a basic entangling operation which can be realised with two laser pulses that address the ions individually (see . also 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 transitions. Short, medium, and long arrows are red sideband, carrier, and blue sideband transitions, respectively. As before, |Si and |Di stand for |gi and |ei in the text and refer to the example of a Ca+ ion.

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

R(1)( π , 0) 1 R(2)(π, 0) 1 |g, g, 0i + 2 √ (|g, g, 0i + i|e, g, 1i) − √ (|g, g, 0i − |e, e, 0i) . −→ 2 −→ 2 √ The pulse sequence creates an entangled state (|g, gi − |e, ei)/ 2 of the two internal qubits while leaving the motional qubit in the ground state. Several specific gate protocols within the two-ion CS have been proposed and implemented. The classic Cirac-Zoller gate works in the following way [7]: first the state . of 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 the motional qubit state is swapped back into ion 1, which completes the gate between the two 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 per vibrational mode, always serves only as a ”bus” connecting the internal qubits of the ions. In particular, the motional qubits cannot be measured independently. Thus at the end of a logic operation, the motional state has to factorise from the state of the internal qubits, which itself can be entangled. . 3 5. State detection. – At the end of a quantum logical process the state of the qubits has to be determined. This is accomplished by exciting the ions and recording the scattered 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 fluorescence or no photons, corresponding to either |gi or |ei. For finding out the probabilities that the ion is in either of the two states after the logic operation, one has to repeat the 12 Jurgen¨ Eschner

Fig. 7. – Creation of laser pulses. Continuous light from a stable laser (coherence time gate time) is sent through an acousto-optical modulator (AOM), which receives a radio frequency signal from a stable oscillator (RF). The RF frequency ∆ is added to the light frequency, thus defining the laser detuning, the RF phase φ translates directly into the phase of the light, and the RF power P controls the light intensity I, including switching it on and off. The AOM process 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 of Fig. 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 is required, during which phase-perturbing or decay processes must be negligible. Apart from stable atomic states, this requires the control of environmental magnetic fields, and in particular it puts challenging requirements on the stability of the laser sources. In the example of the optical qubit, where spontaneous decay of the excited state happens on the 1 s time scale, the laser phase fluctuations are critical. A residual laser linewidth in the 100 Hz range or better is desired in order to maintain phase coherence over a significant number of pulses. To achieve this, sophisticated stabilisation techniques like those developed in the context of optical clocks have to be applied. With hyperfine qubits, laser stability is still very important but less critical, because there the difference frequency between two light fields is relevant, which is in the GHz range and can be made very stable using microwave oscillators. Once the phase-stable light field for driving the qubits is generated, the creation of laser pulses of desired amplitude, detuning, phase shift, and duration is comparatively simple. It is illustrated in Fig. 7. . 4 2. Addressing individual ions in a string. – The conventional way to manipulate individual 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 when the laser beam is scanned across it (unpublished data from Innsbruck). Both axes are linear scales. The data points are ﬁtted by the sum of three Gaussian functions. From the individual Gaussians of about 2.5 µm width (shown as thin lines) one ﬁnds that the intensity on an adjacent ion 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 which can be directed at any ion in the string. This kind of addressing was first demonstrated in [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 the diffraction limit of the focussing lens. This limits the range of possible trap frequencies and thus the speed of gate operations [35]. With the 729 nm light exciting the optical qubit in Ca+, focusing to ∼ 2.5 µm can typically be achieved, leading to an ion spacing around 5 µm. Fig. 8 shows an example. The Rabi frequency on a neighbouring ion can reach about 5% of that on the addressed ion, introducing addressing errors in the quantum logic operations. Such systematic errors could, however, be compensated for by adjustments to pulse lengths and phases [36]. Addressing in gate operations is not indispensable, since protocols have been developed [37] and demonstrated [38, 39] which employ simultaneous laser interaction with both participating ions. The gate used by the NIST group in their recent experiments relies on the action of spin-dependent dipole forces on both ions when they are simultaneously excited [40]. Still, in general a process is required which allows for shielding one ion from the light which excites the adjacent one. The strategy which appears to be . most 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 each ion in the string. At the same time, state read-out should happen fast to allow for a high experimental cycle frequency and to avoid state changes by decay processes. Thus high photon detection efficiency at low dark count rate is desired. A photon counting 14 Jurgen¨ Eschner

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

camera resolving the individual ions is the appropriate device. The suitable parameters are found by acquiring a histogram of photon counts corresponding to the positions of the ions on the camera image and to the two possible signal levels. The histogram will show Poisson distributions around the mean numbers of fluorescence and dark counts, and efficient state discrimination is achieved when the distributions do not overlap. As shown 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 using segmented traps for measuring their final states individually [18].

. 4 4. Problems and solutions. – Some additional experimental tricks are mentioned in this 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, 1i to |e, 2i on the blue and |e, 1i to |g, 2i on the red sideband. Moreover the Rabi√ frequency on the |1i ↔ |2i transitions is larger than on |0i ↔ |1i by a factor of 2. This problem can be solved by composite pulses. This technique is well-known in NMR spectroscopy [42] and its application for ion traps has been proposed in Ref. [43]. An application is the composite SWAP 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|gi + b|ei)(c|1i + d|0i) → (c|gi + d|ei)(a|1i + b|0i) .

It can be obtained by a π-pulse on the blue sideband, but only if the initial state is not |g, 1i. A composite pulse sequence, √ √ √ R = R (π/ 2, 0) R (π 2, φ ) R (π/ 2, 0) , SW AP + √+ SW AP + −1 2 where φSW AP = cos (cot (π/ 2)) ≈ 0.303π , performs the SWAP operation for all states of the CS, by acting as a π-pulse on the |g, 0i ↔ |e, 1i transition and simultaneously as a 4π rotation on |g, 1i ↔ |e, 2i. Composite pulses 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 the state of one ion while the other ions are in superposition states. Therefore these ions have to be protected from the resonant light. This is achieved either by separating the ions 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 couple to the resonant light on the neighbouring ion, to a Zeeman sublevel of D5/2 which is different from the qubit level. After the measurement pulse on the adjacent ion, a −π-pulse reverses the hiding process and restores the superposition of the qubit. Further examples for techniques which deal with experimental imperfections are spin echo pulses to counteract the effect of magnetic field fluctuations and gradients [18], and the compensation of Stark shifts which occur during sideband excitation due to non-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 a hyperfine [40] and in Innsbruck with an optical qubit [34]. The optical implementation with Ca+ ions was very close to the original Cirac-Zoller proposal, while the hyperfine version with Be+ used a protocol based on state-dependent forces, similar to the scheme proposed by Sørensen and Mølmer [37]. Figure 10 displays the result of the Innsbruck experiment. A spectacular result in 2004 was the first deterministic quantum teleportation of the state 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]. The NIST experiment involved shuttling ions between zones of a segmented trap, in order to 16 Jurgen¨ Eschner

Fig. 10. – Experimental implementation of the Cirac-Zoller two-ion quantum CNOT gate with Ca+, from [34]. Each pair of plots shows for both ions the time evolution of the populations of the qubit levels. The initial states are the four basis states, |SSi, |SDi, |DSi, and |DDi. With initial state |Si in ion 1, the state of ion 2 remains the same (left side), whereas initial state |Di in ion 1 causes ion 2 to change its state (right side). Each circle is an average of 100 measurements, where the gate pulse sequence was interrupted at that time and a state measurement was applied to both ions. Each shaded area is a laser pulse or pulse sequence (see top right), starting with the preparation of both ions, then the SWAP operation between ion 1 and the motional qubit; the central pulse sequence on ion 2 is the CNOT gate between its state and the motion, using composite pulses; then the SWAP is reversed on ion 1, which completes the gate.

allow for both individual and simultaneous laser manipulation of the ions. This can be considered as a first step towards scalable trap architecture. The first realisation of a decoherence-free subspace was reported in Ref. [39]. More recently, very long coherence times, of many seconds, of entangled states and states in decoherence-free subspaces have been demonstrated in Refs. [48] and [49]. Several basic quantum logical operations were also demonstrated, such as quantum dense 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 been deterministically 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 detail in A. Steane’s contribution to this volume [56]. The scheme he presents of an ion trap quantum computer which implements full error correction appears to be an overwhelming task. In this final section I summarise a few recent ideas and results which are expected to contribute to meeting this challenge. Three main directions are considered, ion shuttling in segmented traps, sympathetic cooling, and fast gates. Furthermore, I discuss some issues concerning the choice of the optimal qubit. . 6 1. Trap architecture. – Addressing ions in a static string by steering a laser beam becomes technically difficult as the number of ions increases. Moreover, the conventional two-ion gate protocol employing a vibrational mode of a string becomes difficult to implement as the size of the string grows, because the increasing mass of the ion crystal (entering into the Lamb-Dicke parameter η) reduces the coupling on the sideband transitions. Therefore a trap architecture seems favourable which allows for moving the ions. In this scenario, ions are picked from a register and shuttled by time-dependent trap voltages into an interaction zone where two-ion gates are carried out of the kind used in [18]. A detailed scheme has been proposed by the NIST group [57], and several other 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 reported 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 new means of cooling the ions without perturbing their quantum state, i.e. without scattering photons from them. This can be achieved by sympathetic cooling, when the collective modes of a string are cooled by scattering photons only from ions which do not participate in the quantum logic process. Sympathetic Doppler cooling was demonstrated some time ago with Hg+-Be+ mixtures [64] and with Mg+ [65]. Ground state cooling of a two-ion crystal through one of the ions was reported in [32]. Further relevant results were obtained with mixtures of Cd+ isotopes [63] and with a Be+-Mg+ crystal which was cooled to the motional ground state [66]. Very recently, sympathetic cooling and quantum logic have been combined in an experiment with a Be+-Al+ ion crystal, where high-precision spectroscopy was carried out on the Al+ ion, while cooling and state read-out were done on the Be+ ion [67]. . 6 3. Fast gates. – Another major contribution to scaling up ion trap QIP is expected from faster gate operations. The current limitation is the use of spectrally resolved sideband resonances, which requires individual laser pulses to be longer than the inverse trap frequency. Smaller traps with larger frequencies can improve the situation, but 18 Jurgen¨ 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 to excite one mode, that they impart photon momentum kicks and thus interact with a superposition of many modes or even with the local modes of one ion, i.e. its local vibration as if all other ions were fixed [70]. It will be interesting to see how ideas of quantum coherent control can help advancing into this direction [71]. . 6 4. Qubits. – Many issues have to be considered to identify the optimum type of qubit and the optimum ion species. Using an optical qubit in an ion without hyperfine structure makes 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-ion quantum states, which form a decoherence-free subspace [49]. With hyperfine qubits the effect of laser phase noise is reduced because the two fields for the Raman transition are derived from the same source, and their difference frequency is stabilised to a microwave oscillator. The magnetic field sensitivity can also be significantly decreased by using m = 0 magnetic sublevels [72], or by applying a static magnetic field at which the qubit levels have the same differential Zeeman shift [48]. The limitation by spontaneous scattering during a Raman excitation pulse has been assessed in [28], pointing at heavy ions like Cd+ or Hg+ as good candidates. From the practical point of view, the hyperfine qubit in 43Ca+ is an attractive choice for the accessibility of the Ca+ transitions by laser sources. Work with 43Ca+ has been started in Innsbruck and Oxford [73]. Direct excitation of hyperfine qubits by microwave radiation has also been considered [74]. It requires the presence of high magnetic field gradients to obtain sufficient sideband coupling (through an effective Lamb-Dicke parameter), which at the same time would split 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 this final section. The quantum interaction between single trapped ions and single photons is a central objective for many schemes connected to QIP with ions, such as quantum networking [75], distributed quantum computing, or quantum communication. Coherent coupling of ionic qubits to photonic modes may be achieved in a high-finesse resonator. Entangled ion-photon states and entangled states of distant ions can also be generated by projective state preparation conditioned on photon detection events. I review briefly the existing approaches and results with ion traps. Important and highly advanced work in this field is also carried out with trapped neutral atoms [76]. The direct coupling of a trapped ion to a high-finesse optical cavity has been realised in 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 distant trapped ions (after [83]). After preparation of both ions in a state |0i, a laser pulse excites a state-changing transition |0i → |1i with small probability , thus producing the state |0, 0i + (|0, 1i + |1, 0i) + 2|1, 1i. Part of the scattered light from both ions is coherently superimposed on a beam splitter. A photon detection event behind the beam splitter (of π polarisation in the example) does not allow to determine which√ ion changed its state, and therefore signals creation of the entangled state (|0, 1i + |1, 0i)/ 2.

In Innsbruck, Rabi oscillations on a qubit transition induced by a coherent cavity ﬁeld were observed, and cavity sideband excitation was demonstrated [80, 81]. The entanglement between the polarisation of an ion and an emitted photon has been demonstrated in an experiment of the Michigan group [82]. Based on this result, and following proposals by Cabrillo and others [83], several experiments pursue the postselective creation of entangled states of distant ions. It relies on the indistinguishability of photon 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 exciting additional aspect of the future perspectives of ion trap quantum computing. 20 Jurgen¨ Eschner

∗ ∗ ∗ I would like to thank the directors and organisers of the International School of Physics ”Enrico Fermi”, G. Casati, D. L. Shepelyansky, P. Zoller, and G. Benenti for giving me the opportunity to present these lectures. I thank Christian Roos for very helpful 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 in Innsbruck is supported by the Austrian Science Fund (FWF, special research center SFB 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 the Institut f¨ur Quanteninformation GmbH.

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