This thesis describes theoretical and experimental work whose main aim is the development of techniques for using trapped 43Ca⁺ ions for quantum information processing. I present a rate equations model of 43Ca⁺, and compare it with experimental data. The model is then used to investigate and optimise an electron-shelving readout method from a ground-level hyperfine qubit. The process is robust against common experimental imperfections. A shelving fidelity of up to 99.97% is theoretically possible, taking 100 μs. The laser pulse sequence can be greatly simplified for only a small reduction in the fidelity. The simplified method is tested experimentally with fidelities up to 99.8%. The shelving procedure could be applied to other commonly-used species of ion qubit. An entangling two-qubit quantum controlled-phase gate was attempted between a 40Ca⁺ and a 43Ca⁺ ion. The experiment did not succeed due to frequent decrystallisation of the ion pair, and strong motional decoherence. The source of the problems was never identified despite significant experimental effort, and the decision was made to suspend the experiments and continue them in an improved ion trap which is under construction. A sequence of pi-pulses, inspired by the Hahn spin-echo, was derived that is capable of greatly reducing dephasing of any qubit. If the qubit precession frequency varies with time as an nth-order polynomial, an (n+1) pulse sequence is theoretically capable of perfectly cancelling the resulting phase error. The sequence is used on a 43Ca+ magnetic-field-sensitive hyperfine qubit, with 20 pulses increasing the coherence time by a factor of 75 compared to an experiment without any spin-echo. In our ambient noise environment the well-known Carr-Purcell-Meiboom-Gill dynamic-decoupling method was found to be comparably effective.
This thesis describes theoretical and experimental work whose main aim is the development of techniques for using trapped 43Ca⁺ ions for quantum information processing. I present a rate equations model of 43Ca⁺, and compare it with experimental data. The model is then used to investigate and optimise an electron-shelving readout method from a ground-level hyperfine qubit. The process is robust against common experimental imperfections. A shelving fidelity of up to 99.97% is theoretically possible, taking 100 μs. The laser pulse sequence can be greatly simplified for only a small reduction in the fidelity. The simplified method is tested experimentally with fidelities up to 99.8%. The shelving procedure could be applied to other commonly-used species of ion qubit. An entangling two-qubit quantum controlled-phase gate was attempted between a 40Ca⁺ and a 43Ca⁺ ion. The experiment did not succeed due to frequent decrystallisation of the ion pair, and strong motional decoherence. The source of the problems was never identified despite significant experimental effort, and the decision was made to suspend the experiments and continue them in an improved ion trap which is under construction. A sequence of pi-pulses, inspired by the Hahn spin-echo, was derived that is capable of greatly reducing dephasing of any qubit. If the qubit precession frequency varies with time as an nth-order polynomial, an (n+1) pulse sequence is theoretically capable of perfectly cancelling the resulting phase error. The sequence is used on a 43Ca+ magnetic-field-sensitive hyperfine qubit, with 20 pulses increasing the coherence time by a factor of 75 compared to an experiment without any spin-echo. In our ambient noise environment the well-known Carr-Purcell-Meiboom-Gill dynamic-decoupling method was found to be comparably effective.
This thesis is concerned with the development of an intermediate magnetic field "clock-qubit" in 43Ca+ at 146G and techniques to manipulate this qubit using microwaves and lasers. While 43Ca+ has previously been used as a qubit, its relatively complicated level structure - with a nuclear spin of 7/2 and low-lying D-states -- makes cooling it in the intermediate field an intimidating prospect. As a result, previous experiments have used small magnetic fields of a few gauss where coherence times are limited and off-resonant excitation is a significant source of experimental error. We demonstrate a simple scheme that allows 43Ca+ to be cooled in the intermediate field without any additional experimental complexity compared with low fields. Using the clock-qubit, we achieve a coherence time of T*2 = 50 (10)s - the longest demonstrated in any single qubit. We also demonstrate a combined state preparation and measurement error of 6.8(6)x 10-4 - the lowest achieved for a hyperfine trapped ion qubit [NVG+13] - and single-qubit logic gates with average errors of 1.0(3) x 10-6 - more than an order of magnitude better than the previous record [BWC+11]. These results represent the state-of-the-art in the field of single-qubit control. Moreover, we achieve them all in a single scalable room-temperature ion trap using experimentally robust techniques and without relying on the use of narrow-linewidth lasers, magnetic field screening or dynamical decoupling techniques. We also present work on a recent scheme [OWC+11] to drive two-qubit gates using microwaves. We have constructed an ion trap with integrated microwave circuitry to perform these gates. Using this trap, we have driven motional sideband transitions, demonstrating the spin-motion coupling that underlies the two-qubit gate. We present an analysis of likely sources of experimental error during a future two-qubit gate and the design and preliminary characterisation of apparatus to minimise the main error contributions. Using this apparatus, we hope to perform a two-qubit gate in the near future.
Quantum computers offer great potential for significant speedup in executing certain algorithms compared to their classical counterparts. One of the most promising physical systems in which implementing such a device seems viable are trapped atomic ions. All of the fundamental operations needed for quantum information processing have already been experimentally demonstrated in trapped ion systems. Today, the remaining two obstacles are to improve the fidelities of these operations up to the point where quantum error correction techniques can be successfully applied, as well as to scale up the present systems to a higher number of quantum bits (qubits). This thesis addresses both issues. On the one hand, it decribes the experimental implementation of a high-fidelity two-qubit quantum logic gate, which is the most technically demanding fundamental operation to realise in practice. On the other hand, the presented work is carried out in a microfabricated surface ion trap – an architecture that holds the promise of scalability. The gate is applied directly to hyperfine "atomic clock" qubits in 43Ca+ ions using the near-field microwave magnetic field gradient produced by an integrated trap electrode. To protect the gate against fluctuating energy shifts of the qubit states, as well as to avoid the need to null the microwave field at the position of the ions, a dynamically decoupled Mølmer-Sørensen scheme is employed. After accounting for state preparation and measurement errors, the achieved gate fidelity is 99.7(1)%. In previous work, the same apparatus has been used to demonstrate coherence times of T*2 ≈ 50 s and all single-qubit operations with fidelity > 99.95%. To gain access to the "atomic clock" qubit transition in 43Ca+, a static magnetic field of 146G is applied. The resulting energy level Zeeman-structure is spread over many times the linewidth of the atomic transition used for Doppler cooling. This thesis presents a simple and robust method for Doppler cooling and obtaining high fluorescence from this qubit in spite of the complicated level structure. A temperature of 0.3mK, slightly below the Doppler limit, is reached.
We demonstrate the use of dynamic decoupling techniques to extend the coherence time of a single memory qubit by nearly two orders of magnitude. By extending the Hahn spin-echo technique to correct for unknown, arbitrary polynomial variations in the qubit precession frequency, we show analytically that the required sequence of π-pulses is identical to the Uhrig dynamic decoupling (UDD) sequence. We compare UDD and Carr–Purcell–Meiboom–Gill (CPMG) sequences applied to a single 43 Ca + trapped-ion qubit and find that they afford comparable protection in our ambient noise environment.
We report on experiments with strings of laser-cooled Ca+ ions for quantum information processing. An experiment using the isotope 43Ca+ was set up from scratch. We devised techniques for loading and laser-cooling 43Ca+ ion strings, initializing the ions in the hyperfine clock states and reading out the quantum state with high efficiency. Coherence times of the qubits are about 1 s. The single-qubit gates were implemented using microwave and Raman excitation, and coupling to the motional state was demonstrated. Very recently, a pair of 43Ca+ ions were entangled for the first time. In a second line of experiments, we demonstrated simple quantum algorithms with few 40Ca+ ions as well as the implementation of a high-fidelity gate operation entangling a pair of ions with a fidelity of 99.3(1)%. Among the milestone experiments are the demonstration and quantum process tomography of a Cirac-Zoller gate with mean fidelity of about 93%, the demonstration of entanglement in a decoherence-free sub space lasting for 20s, the demonstration of deterministic quantum teleportation, a Toffoli gate, deterministic entanglement swapping a partial measurement of a GHZ state preserving entanglement in the remaining qubits, creation and analysis of an 8-ion W-state, the application of entangled ions for precision spectroscopy.
Quantum entangling logic gates are key ingredients for the implementation of a quantum information processing device. In this thesis, we focus on experimental implementations of three types of entangling geometric-phase gates with trapped ions, which rely on the effective spin-spin interactions generated with state-dependent forces. First, a mixed-species entangling gate is demonstrated using a 9Be+ and a 25Mg+ ion to create a Bell state with a fidelity of 0.979(1). Combined with single-qubit gates, we use this mixed-species gate to implement controlled-NOT and SWAP gates. Second, we implement a high-fidelity universal gate set with 9Be+ ions. Single-qubit gates with error per gate of 3.8(1)x10-5 are achieved. By creating a Bell state with a deterministic two-qubit entangling gate, we deduce a gate error as low as 8(4)x10-4. Third, a novel two-qubit entangling gate with dynamical decoupling built-in is demonstrated with a fidelity of 0.974(4). This gate is robust against qubit dephasing errors and offers simplifications in experimental implementation compared to some other gates with trapped ions. Errors in the above implementations are evaluated and methods to further reduce imperfections are discussed. In a separate experiment, correlated measurements made on pairs of ions violate a "chained" Bell inequality obeyed by any local-realistic theory. The lowest chained Bell inequality parameter determined from our measurements is 0.296(12), this value is significantly lower than 0.586, the minimum value derived from a perfect Clauser-Horne-Shimony-Horne (CHSH) Bell inequality experiment. Furthermore, our CHSH Bell inequality results provide a device-independent certification of the deterministically created Bell states.
Over the last decade quantum information processing (QIP) has exploded into a major field of physics, studied both experimentally and theoretically with a universal quantum computer as a long-term vision. Ions stored as strings in linear Paul traps are among the most promising systems for constructing a quantum device harnessing the computing power inherent in the laws of quantum physics. The two most important challenges in trapped ion quantum computing today are to realize and control systems with large numbers of ions, and to improve and integrate the operations which serve as basic building blocks, in particular universal gate operations. Quite surprisingly, even after many years of intense research with different atom species, it is still not evident which ion is best suited for QIP. This thesis describes the construction of a new experiment built to study a promising candidate qubit, namely the calcium isotope 43Ca+. Due to its nuclear spin of I = 7/2, the isotope 43Ca+ exhibits a relatively complicated level structure, compared to other ions used so far for QIP. This work develops various strategies to successfully control such a complex level scheme. Initially we investigated the hyperfine structure of the 4s S1/2 ↔ 3d D5/2 quadrupole transition at a wavelength of 729 nm by laser spectroscopy using a single trapped 43Ca+ ion. We determined the hyperfine structure constants of the metastable level as AD5/2 = −3.893 12(3)MHz and BD5/2 = −4.239(1)MHz. The isotope shift of this transition with respect to 40Ca+ was measured to be ∆ iso /(2π) = 4 134 711 720(390)Hz. Armed with this knowledge, we were able to demonstrate ground-state cooling, robust state initialization and efficient readout of a qubit encoded in the ground-state hyperfine structure of a 43Ca+ ion. A microwave and a Raman light field were used to drive qubit transitions, and the coherence times for both fields were compared. Phase errors due to interferometric instabilities in the Raman field generation were not limiting the experiments on a time scale of at least 100ms. Even in the presence of magnetic field fluctuations we found a quantum information storage time exceeding one second. We implemented a Molmer-Sorensen type gate operation, entangling ions with a fidelity of 99.3(1)%. The gate was performed on a pair of qubits encoded in two trapped 40Ca+ ions using an amplitude-modulated laser beam interacting with both ions at the same time. A robust gate operation, mapping separable states onto maximally entangled states was achieved by adiabatically switching the laser-ion coupling on and off. The performance of a single gate and concatenations of up to 21 gate operations were analyzed. Concerning the fidelity, this result sets a world record for creating two-qubit entanglement on-demand irrespective of the physical realization considered. Zusammenfassung Die Quanteninformationsverarbeitung (QIV) als experimentelles und theoretisches Teilgebiet der Physik mit dem universellen Quantencomputer als Fernziel entwickelte sich innerhalb der letzten Dekade geradezu explosionsartig. Fur die technische Realisierung einer Maschine zur Ausnutzung der den Gesetzen der Quantenmechanik eigenen Rechenkraft gibt es eine Vielzahl an Moglichkeiten. Ionen, die sich in linearen Paul-Fallen wie auf einer Perlenschnur aufreihen, haben sich dabei mittlerweile als sehr vielversprechenden Ansatz etabliert. Die beiden grosten Herausforderungen der Ionenfallen-Experimente zur QIV bestehen gegenwartig in der Skalierung zu einer Vielzahl kontrollierter Quantenbits und in der Verbesserung und Integration der bereits demonstrierten Grundbausteine, insbesondere universeller Gatteroperationen. Uberraschenderweise ist auch nach vielen Jahren intensiver Forschung unklar, welche Ionen-Sorte zu diesem Zweck am besten geeignet ist. Die vorliegende Arbeit beschreibt den Aufbau eines neuen Experiments zur Untersuchung des Calciumisotops 43Ca+ als attraktiven Kandidaten. Aufgrund des Kernspins von I = 7/2 hat das Isotop 43Ca+, verglichen mit Atomsorten, die bisher fur die QIV verwandt wurden, ein komplizierteres Energieniveauschema. Diese Arbeit diskutiert die daraus resultierenden Moglichkeiten und beschreibt deren experimentelle Realisierung, so dass diese Komplexitat erfolgreich genutzt werden kann. Zunachst wurde die Hyperfein-Struktur des 4s S1/2 ↔ 3d D5/2 Quadrupol-Ubergangs bei einer Wellenlange von 729 nm mittels Laserspektroskopie an einem einzelnen 43Ca+Ion untersucht. Die Hyperfein-Konstanten des metastabilen Energieniveaus wurden zu AD5/2 = −3.893 12(3)MHz, BD5/2 = −4.239(1)MHz bestimmt. Als Nebenprodukt ergab sich die Isotopieverschiebung auf diesem Ubergang bezogen auf 40Ca+ zu ∆ iso /(2π) = 4 134 711 720(390)Hz. Mit diesem Wissen gelang es einzelne 43Ca+-Ionen nahe an den Bewegungsgrundzustand einer Mode zu kuhlen, sowie 43Ca+-Hyperfein-Qubits mit hoher Gute zu initialisieren und auszulesen. Die Koharenzzeiten dieses Hyperfein-Qubits wurden mit einem Mikrowellenfeld und einem Raman-Lichtfeld gemessen. Trotz der Anwesenheit magnetischer Storfelder konnte Quanteninformation uber eine Sekunde gespeichert werden. Auserdem ergaben Messungen, dass die interferometrischen Stabilitatsanforderungen der Raman-Lichtfelder, auf einer Zeitskala von 100ms keine Limitierung darstellen. Die Arbeit schliest mit der Beschreibung einer Molmer-Sorensen Gatteroperation, die verschrankte Zustande mit einer Gute von 99.3(1)% erzeugte. Dieser Gattertyp wurde erstmals auf einem optischen Qubit mittels eines amplitudenmodulierten Lasers realisiert, der an zwei 40Ca+-Ionen gleichermasen koppelte. Eine robuste Gatteroperation, die separable in verschrankte Zustande uberfuhrt, wurde durch adiabatisches Anund Ausschalten dieses Lasers erreicht. Zur Bestimmung moglicher Fehlerquellen wurden bis zu 21 aneinandergereihte Gatteroperationen analysiert. Bezuglich der erreichten Gute stellt das Ergebnis die momentan beste deterministische Zwei-Qubit-Verschrankungsoperation aller bekannter Systeme dar.