A. M. Steane

Robust qubit memory is essential for quantum computing, both for near-term devices operating without error correction, and for the long-term goal of a fault-tolerant processor. We directly measure the memory error $ε_m$ for a $^{43}$Ca$^+$ trapped-ion qubit in the small-error regime and find $ε_m<10^{-4}$ for storage times $t\lesssim50\,\mbox{ms}$. This exceeds gate or measurement times by three orders of magnitude. Using randomized benchmarking, at $t=1\,\mbox{ms}$ we measure $ε_m=1.2(7)\times10^{-6}$, around ten times smaller than that extrapolated from the $T_{2}^{\ast}$ time, and limited by instability of the atomic clock reference used to benchmark the qubit.

Quantum bits based on individual trapped atomic ions constitute a promising technology for building a quantum computer, with all the elementary operations having been achieved with the necessary precision for some error-correction schemes. However, the essential two-qubit logic gate used for generating quantum entanglement has hitherto always been performed in an adiabatic regime, where the gate is slow compared with the characteristic motional frequencies of ions in the trap, giving logic speeds of order 10kHz. There have been numerous proposals for performing gates faster than this natural "speed limit" of the trap. We implement the method of Steane et al., which uses tailored laser pulses: these are shaped on 10 ns timescales to drive the ions' motion along trajectories designed such that the gate operation is insensitive to optical phase fluctuations. This permits fast (MHz-rate) quantum logic which is robust to this important source of experimental error. We demonstrate entanglement generation for gate times as short as 480ns; this is less than a single oscillation period of an ion in the trap, and 8 orders of magnitude shorter than the memory coherence time measured in similar calcium-43 hyperfine qubits. The method's power is most evident at intermediate timescales, where it yields a gate error more than ten times lower than conventional techniques; for example, we achieve a 1.6 us gate with fidelity 99.8%. Still faster gates are possible at the price of higher laser intensity. The method requires only a single amplitude-shaped pulse and one pair of beams derived from a continuous-wave laser, and offers the prospect of combining the unrivalled coherence properties, operation fidelities and optical connectivity of trapped-ion qubits with the sub-microsecond logic speeds usually associated with solid state devices.

We describe a simple approach to the problem of incorporating the response time of an atom or ion being Doppler-cooled into the theory of the cooling process. The system being cooled does not in general respond instantly to the changing laser frequencies it experiences in its rest frame, and this "dynamic effect" can affect significantly the temperatures attainable. It is particularly important for trapped ions when there is a slow decay out of the cooling cycle requiring the use of a repumping beam. We treat the cases of trapped ions with two and three internal states, then apply the theory to $^{40}{\rm Ca}^+$. For this ion experimental data exist showing the ion to be cold under conditions for which heating is predicted if the dynamic effect is neglected. The present theory accounts for the observed behaviour.

We demonstrate simple and robust methods for Doppler cooling and obtaining high fluorescence from trapped 43Ca+ ions at a magnetic field of 146 Gauss. This field gives access to a magnetic-field-independent "atomic clock" qubit transition within the ground level hyperfine structure of the ion, but also causes the complex internal structure of the 64 states relevant to Doppler cooling to be spread over many times the atomic transition line-width. Using a time-dependent optical Bloch equation simulation of the system we develop a simple scheme to Doppler-cool the ion on a two-photon dark resonance, which is robust to typical experimental variations in laser intensities, detunings and polarizations. We experimentally demonstrate cooling to a temperature of 0.3 mK, slightly below the Doppler limit for the corresponding two-level system, and then use Raman sideband laser cooling to cool further to the ground states of the ion's radial motional modes. These methods will enable two-qubit entangling gates with this ion, which is one of the most promising qubits so far developed.

Entanglement is one of the most fundamental properties of quantum mechanics, and is the key resource for quantum information processing. Bipartite entangled states of identical particles have been generated and studied in several experiments, and post-selected or heralded entangled states involving pairs of photons, single photons and single atoms, or different nuclei in the solid state, have also been produced. Here, we use a deterministic quantum logic gate to generate a "hybrid" entangled state of two trapped-ion qubits held in different isotopes of calcium, perform full tomography of the state produced, and make a test of Bell's inequality with non-identical atoms. We use a laser-driven two-qubit gate, whose mechanism is insensitive to the qubits' energy splittings, to produce a maximally-entangled state of one Ca-40 qubit and one Ca-43 qubit, held 3.5 microns apart in the same ion trap, with 99.8(6)% fidelity. We test the Clauser-Horne-Shimony-Holt (CHSH) version of Bell's inequality for this novel entangled state and find that it is violated by 15 standard deviations; in this test, we close the detection loophole but not the locality loophole. Mixed-species quantum logic is a powerful technique for the construction of a quantum computer based on trapped ions, as it allows protection of memory qubits while other qubits undergo logic operations, or are used as photonic interfaces to other processing units. The entangling gate mechanism used here can also be applied to qubits stored in different atomic elements; this would allow both memory and logic gate errors due to photon scattering to be reduced below the levels required for fault-tolerant quantum error correction, which is an essential pre-requisite for general-purpose quantum computing.

We show how to create quantum gates of arbitrary speed between trapped ions, using a laser walking wave, with complete insensitivity to drift of the optical phase, and requiring cooling only to the Lamb-Dicke regime. We present pulse sequences that satisfy the requirements and are easy to produce in the laboratory.

We propose a surface ion trap design incorporating microwave control electrodes for near-field single-qubit control. The electrodes are arranged so as to provide arbitrary frequency, amplitude and polarization control of the microwave field in one trap zone, while a similar set of electrodes is used to null the residual microwave field in a neighbouring zone. The geometry is chosen to reduce the residual field to the 0.5% level without nulling fields; with nulling, the crosstalk may be kept close to the 0.01% level for realistic microwave amplitude and phase drift. Using standard photolithography and electroplating techniques, we have fabricated a proof-of-principle electrode array with two trapping zones. We discuss requirements for the microwave drive system and prospects for scalability to a large two-dimensional trap array.

We describe and implement a method to restore the state of a single qubit, in principle perfectly, after it has partially collapsed. The method resembles the classical Hahn spin-echo, but works on a wider class of relaxation processes, in which the quantum state partially leaves the computational Hilbert space. It is not guaranteed to work every time, but successful outcomes are heralded. We demonstrate using a single trapped ion better performance from this recovery method than can be obtained employing projection and post-selection alone. The demonstration features a novel qubit implementation that permits both partial collapse and coherent manipulations with high fidelity.

We demonstrate a Doppler cooling and detection scheme for ions with low-lying D levels which almost entirely suppresses scattered laser light background, while retaining a high fluorescence signal and efficient cooling. We cool a single ion with a laser on the 2S1/2 to 2P1/2 transition as usual, but repump via the 2P3/2 level. By filtering out light on the cooling transition and detecting only the fluorescence from the 2P_3/2 to 2S1/2 decays, we suppress the scattered laser light background count rate to 1 per second while maintaining a signal of 29000 per second with moderate saturation of the cooling transition. This scheme will be particularly useful for experiments where ions are trapped in close proximity to surfaces, such as the trap electrodes in microfabricated ion traps, which leads to high background scatter from the cooling beam.

Laser-cleaning of the electrodes in a planar micro-fabricated ion trap has been attempted using ns pulses from a tripled Nd:YAG laser at 355nm. The effect of the laser pulses at several energy density levels has been tested by measuring the heating rate of a single 40Ca+ trapped ion as a function of its secular frequency. A reduction of the electric-field noise spectral density by ~50% has been observed and a change in the frequency dependence also noticed. This is the first reported experiment where the "anomalous heating" phenomenon has been reduced by removing the source as opposed to reducing its thermal driving by cryogenic cooling. This technique may open the way to better control of the electrode surface quality in ion microtraps.

We characterise the performance of a surface-electrode ion "chip" trap fabricated using established semiconductor integrated circuit and micro-electro-mechanical-system (MEMS) microfabrication processes which are in principle scalable to much larger ion trap arrays, as proposed for implementing ion trap quantum information processing. We measure rf ion micromotion parallel and perpendicular to the plane of the trap electrodes, and find that on-package capacitors reduce this to <~ 10 nm in amplitude. We also measure ion trapping lifetime, charging effects due to laser light incident on the trap electrodes, and the heating rate for a single trapped ion. The performance of this trap is found to be comparable with others of the same size scale.

Currently laser cooling schemes are fundamentally based on the weak coupling regime. This requirement sets the trap frequency as an upper bound to the cooling rate. In this work we present a numerical study that shows the feasibility of cooling in the strong coupling regime which then allows cooling rates that are faster than the trap frequency with state of the art experimental parameters. The scheme we present can work for trapped atoms or ions as well as mechanical oscillators. It can also cool medium size ions chains close to the ground state.

We describe the fabrication and characterization of a new surface-electrode Paul ion trap designed for experiments in scalable quantum information processing with Ca+. A notable feature is a symmetric electrode pattern which allows rotation of the normal modes of ion motion, yielding efficient Doppler cooling with a single beam parallel to the planar surface. We propose and implement a technique for micromotion compensation in all directions using an infrared repumper laser beam directed into the trap plane. Finally, we employ an alternate repumping scheme that increases ion fluorescence and simplifies heating rate measurements obtained by time-resolved ion fluorescence during Doppler cooling.

We demonstrate sympathetic cooling of a 43Ca+ trapped-ion "memory" qubit by a 40Ca+ "coolant" ion near the ground state of both axial motional modes, whilst maintaining coherence of the qubit. This is an essential ingredient in trapped-ion quantum computers. The isotope shifts are sufficient to suppress decoherence and phase shifts of the memory qubit due to the cooling light which illuminates both ions. We measure the qubit coherence during 10 cycles of sideband cooling, finding a coherence loss of 3.3% per cooling cycle. The natural limit of the method is O(0.01%) infidelity per cooling cycle.

We demonstrate long-lived coherence in internal hyperfine states of a single \Ca{43} trapped-ion qubit $[T_2=1.2(2)\s]$, and in external motional states of a single \Ca{40} trapped-ion qubit $[T_2'=0.18(4)\s]$, in the same apparatus. The motional decoherence rate is consistent with the heating rate, which was measured to be 3(1) quanta/sec. Long coherence times in the external motional states are essential for performing high-fidelity quantum logic gates between trapped-ion qubits. The internal-state $T_2$ time that we observe in \Ca{43}, which has not previously been used as a trapped-ion qubit, is about one thousand times longer than that of physical qubits based on \Ca{40} ions. Using a single spin-echo pulse to ``re-phase'' the internal state, we can detect no decoherence after 1\s, implying an effective coherence time $T_2^{\mbox{\tiny SE}} \gtish 45\s$. This compares with timescales in this trap for single-qubit operations of \ish 1\us, and for two-qubit operations of \ish 10\us.

We create entangled states of the spin and motion of a single $^{40}$Ca$^+$ ion in a linear ion trap. The motional part consists of coherent states of large separation and long coherence time. The states are created by driving the motion using counterpropagating laser beams. We theoretically study and experimentally observe the behaviour outside the Lamb-Dicke regime, where the trajectory in phase space is modified and the coherent states become squeezed. We directly observe the modification of the return time of the trajectory, and infer the squeezing. The mesoscopic entanglement is observed up to $Δα= 5.1$ with coherence time 170 microseconds and mean phonon excitation $\nbar = 16$.

We have implemented a universal quantum logic gate between qubits stored in the spin state of a pair of trapped calcium 40 ions. An initial product state was driven to a maximally entangled state deterministically, with 83% fidelity. We present a general approach to quantum state tomography which achieves good robustness to experimental noise and drift, and use it to measure the spin state of the ions. We find the entanglement of formation is 0.54.

We study the problem of designing electrode structures that allow pairs of ions to be brought together and separated rapidly in an array of linear Paul traps. We show that it is desirable for the electrode structure to produce a d.c. octupole moment with an a.c. radial quadrupole. For the case where electrical breakdown limits the voltages that can be applied, we show that the octupole is more demanding than the quadrupole when the characteristic distance scale of the structure is larger than 1 to 10 microns (for typical materials). We present a variety of approaches and optimizations of structures consisting of one to three layers of electrodes. The three-layer structures allow the fastest operation at given distance r from the trap centres to the nearest electrode surface, but when the total thickness w of the structure is constrained, leading to w < r, then two-layer structures may be preferable.

We present studies of resonance-enhanced photo-ionization for isotope-selective loading of Ca+ into a Paul trap. The 4s^2 ^1S_0 <-> 4s4p ^1P_1 transition of neutral calcium is driven by a 423nm laser and the atoms are photo-ionized by a second laser at 389nm. Isotope-selectivity is achieved by using crossed atomic and laser beams to reduce the Doppler width significantly below the isotope shifts in the 423nm transition. The loading rate of ions into the trap is studied under a range of experimental parameters for the abundant isotope 40Ca+. Using the fluorescence of the atomic beam at 423nm as a measure of the Ca number density, we estimate a lower limit for the absolute photo-ionization cross-section. We achieve loading and laser-cooling of all the naturally occurring isotopes, without the need for enriched sources. Laser-heating/cooling is observed to enhance the isotope-selectivity. In the case of the rare species 43Ca+ and 46Ca+, which have not previously been laser-cooled, the loading is not fully isotope-selective but we show that pure crystals of 43Ca+ may nevertheless be obtained. We find that for loading 40Ca+ the 389nm laser may be replaced by an incoherent source.

We first consider the basic requirements for a quantum computer, arguing for the attractiveness of nuclear spins as information-bearing entities, and light for the coupling which allows quantum gates. We then survey the strengths of and immediate prospects for quantum information processing in ion traps. We discuss decoherence and gate rates in ion traps, comparing methods based on the vibrational motion with a method based on exchange of photons in cavity QED. We then sketch the main features of a quantum computer designed to allow an algorithm needing 10^6 Toffoli gates on 100 logical qubits. We find that around 200 ion traps linked by optical fibres and high-finesse cavities could perform such an algorithm in a week to a month, using components at or near current levels of technology.

We report a precise search for correlation effects in linear chains of 2 and 3 trapped Ca+ ions. Unexplained correlations in photon emission times within a linear chain of trapped ions have been reported, which, if genuine, cast doubt on the potential of an ion trap to realize quantum information processing. We observe quantum jumps from the metastable 3d 2D_{5/2} level for several hours, searching for correlations between the decay times of the different ions. We find no evidence for correlations: the number of quantum jumps with separations of less than 10 ms is consistent with statistics to within errors of 0.05%; the lifetime of the metastable level derived from the data is consistent with that derived from independent single-ion data at the level of the experimental errors 1%; and no rank correlations between the decay times were found with sensitivity to rank correlation coefficients at the level of |R| = 0.024.

We report a measurement of the lifetime of the 3d 2D_{5/2} metastable level in 40Ca+, using quantum jumps of a single cold calcium ion in a linear Paul trap. The 4s S_{1/2} - 3d D_{5/2} transition is significant for single-ion optical frequency standards, astrophysical references, and tests of atomic structure calculations. We obtain tau = 1.168 +- 0.007 s from observation of nearly 64,000 quantum jumps during approximately 32 hours. Our result is more precise and significantly larger than previous measurements. Experiments carried out to quantity systematic effects included a study of a previously unremarked source of systematic error, namely excitation by the broad background of radiation emitted by a semiconductor diode laser. Combining our result with atomic structure calculations yields 1.20 +- 0.01 s for the lifetime of 3d D_{3/2}. We also use quantum jump observations to demonstrate photon anti-bunching, and to estimate background pressure and heating rates in the ion trap.

Two techniques are described that simplify the experimental requirements for measuring and manipulating quantum information stored in trapped ions. The first is a new technique using electron shelving to measure the populations of the Zeeman sublevels of the ground state, in an ion for which no cycling transition exists from any of these sublevels. The second technique is laser cooling to the vibrational ground state, without the need for a trap operating in the Lamb-Dicke limit. This requires sideband cooling in a sub-recoil regime. We present a thorough analysis of sideband cooling on one or a pair of sidebands simultaneously.