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 tech- nical 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 pa- rameters 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 be- fore 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 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 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 ampli- tude 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.

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