Introduction

Background

Simulating quantum systems is known to be a very difficult computational problem. In the early 1980s Richard Feynman suggested that only a quantum system would be able to efficiently simulate another quantum system. This observation led to the development of quantum information processing. Remarkable progress that has been made in this field, yet the original idea of a quantum simulator has not yet been put into practice although some attempts have been made. However, things are about to change due to the proposal by Porras and Cirac (Phys. Rev. Lett. 96:250501 (2006)) for quantum simulation and computation with planar Coulomb crystals. This scheme is particularly promising because it makes use of the ion trap quantum computation technologies that at the moment are the most successful among the proposed systems for implementing quantum information processing. So far, in ion traps all the required building blocks for quantum computation have been proved. The challenge that is left is scalability. Planar Coulomb crystals also hold the potential for scalable ion trap quantum computation. Moreover, it was estimated that even as few as 30-40 ions would be enough to simulate quantum systems that are beyond the computing power of today's top supercomputers.

Motivation and purpose

Motivated by the prospect of the experimental realization of applications like quantum simulation, scalable quantum computation and even measurement-based quantum computation, we have undertaken the task of investigating in detail the capability of planar Coulomb crystals to implement the above. As the next step, we aim at realizing planar Coulomb crystals in experiment, our long-term goal (beyond the scope of the current work) being the implementation of quantum simulation. For fulfilling our goal, the following steps are required: (i) a detailed investigation of planar Coulomb crystals, (ii) the estimation of the efficiency of Porras-Cirac proposal and the analysis of the implementation issues, (iii) the design and construction of an experimental setup optimized for the realization of planar Coulomb crystals.

Study of planar crystals

Simulation methods

Molecular dynamics simulations have long been used in the study of Coulomb crystals in both Penning and RF traps. Our simulations are designed using the ProtoMol framework, which has been already extensively tested for the simulation of single and multi-component Coulomb crystals in RF traps. We ran the large simulations on a 16 node PC-cluster and the smaller ones on a single machine (Core2Duo 2.4GHz 2GB/RAM). All the nodes of the cluster contain an Intel P4 3.2GHz processor with 2GB of memory and are connected through Gigabit ethernet. At each time step the forces are estimated and the equations of motion are integrated using the Leapfrog algorithm. The Coulomb force between the ions and was usually calculated directly and only when the number of ions exceeds 10,000 the Multi-grid algorithm was used. We also introduced collisions with background gas. Other heating sources like patch fluctuations are insufficiently understood, hence, they were not included.

Planar crystals

Trapped, cooled ions crystallize when the coupling parameter is larger than ~173. In the case of infinite plasmas bcc-crystals are expected. However, in experiments the number of ions is not large enough and the crystal structures are determined by the boundary conditions set by the confining potentials. 1D, 2D and 3D structures varying from strings and zig-zags to spheroidal crystals have been observed in experiments. The shape of Coulomb crystals depends on the confining potentials. In the case of harmonic potentials, the crystal shape depends only on the anisotropy parameter which is the ratio between the axial and radial frequencies. An oblate spheroidal crystal with N ions will turn into a planar one when the anisotropy parameter is larger than a certain threshold. In planar crystals the triangular-lattice ordering occurs near the center while at the edge shell structures appear. We derived the shell structure for planar crystals with 100 to 5,000 ions and we also calculated the radius and average distance between the ions for crystals with different numbers of ions up to 10,000. We simulated 10,000 ions planar crystals in both Penning and RF traps.

Heating

RF-heating results from the chaotic motion of the ions and manifests itself as an increase in the kinetic energy. Several studies have considered the effects of RF-heating and it has been shown that at low temperatures RF-heating is quite small. The dependence of the RF-heating rates on the trapping voltages, temperature and number of ions has never before been studied for planar crystals. We studied these in detail and found that larger crystals heat faster and in general the heating rates are discouragingly large meaning that planar crystal may not be maintained without continuous cooling. However, we also found that the heating rate strongly depends on the trapping conditions and can be significantly decreased by choosing the appropriate trapping parameters. In order to find ways of reducing heating we investigated the RF-heating dependence on the trap parameters. We found that in order to obtain reduced RF-heating rates low voltages are required. Furthermore, we investigated the effect of collisions with background gas in both Penning and RF-traps and we found out that, considering the usual experimental conditions, in Penning traps heating due to background gas collisions is very significant while in RF-traps it is very small.

Multi-component planar crystals

Bicrystals (crystals composed of two ion species or two isotopes of the same species) have been studied theoretically and experimentally before, however, there are no results on planar bicrystals. We studied planar bicrystals in the view of utilizing them for quantum simulation. In bicrystals the heavy ions form an outer shell, while the light ions gather at the center. The separation between the outer and inner shell depends on the ratio of the ions' masses. The advantage of bicrystals is that while continuously cooling the outer shell, the inner shell is also cooled by sympathetic cooling and, therefore, the crystal can be maintained for long times. Moreover, since the inner shell is not cooled directly, it can be used for quantum simulation. We investigated planar bicrystals in RF traps and found that because of the mass dependence of the radial potential interesting features occur. First, the number of ions in the inner shell is limited by the mass difference between the two components. We calculated the maximum number of ions in the inner shell as a function of the ion masses. Second, the spatial separation between the shells depends on the trapping parameters (i.e. the radial frequency shift) as well as the ion masses and charges. We derived the spatial separation between the shells when strong axial confinement both theoretically and numerically. Then we investigated the effect of spatial separation on sympathetic cooling and RF-heating.

Quantum simulation and computation with planar crystals

In planar crystals, in certain conditions, the ions moving in the axial direction can be considered as independent harmonic oscillators weakly coupled by the Coulomb interaction. Therefore, a planar crystal is similar to a microtrap array. Two-qubit interactions necessary for the implementation of quantum simulation and computation are realized by the means of a spin-dependent pushing force. There are several ways in which this pushing force can be induced. We studied the "push-gate" (for quantum computation and simulation) and the effective spin-spin interaction (for quantum simulation) and estimated the error as a function of various trap and laser parameters. When implementing the "push-gate" in the axial direction, the anharmonic terms of the Coulomb force induce a coupling between the axial motion and the hot radial vibrational modes which leads to decoherence. However, it was shown that this is not such a big issue and in principle high fidelity quantum gates could be achieved. In order to be able to realize the "push-gate" in a planar crystal it is necessary that the Coulomb energy is small compared to the potential energy. When studying the effects of heating on the fidelity of the "push-gate" the most important parameters are the temperature and the distance between the ions. The heating of the planar crystal results in the fluctuations of these two parameters so we estimated the error dependence in these cases. Moreover, we estimated the error in the implementation of effective spin-spin interactions.

Design and construction of the ion trap system

The necessity of minimizing the effects of heating and the specific requirements for the realization of planar crystals would make experiments difficult in both Penning and RF traps. However, each type of trap has its own merits and demerits. In RF-traps, while large crystals could not be maintained without continuous cooling, smaller crystals (<100 ions) cooled to very low temperatures and with a careful choice of the operating parameters may be possible to realize in experiment. The suppression of heating from other sources would be also required. The RF-heating is an important issue but low temperatures might be maintained by sympathetic cooling. The high vacuum conditions in RF-trap experiments together with the more flexible architecture of the trap electrodes providing very good optical access and lower voltages, as well as the uncomplicated laser cooling are the advantages of RF-traps. For the "proof-of-principle", small crystals in RF-traps may be easier to realize, therefore, we decided to construct such a trap.

The design of an RF-trap specialized for the realization of planar crystals is not by any means a trivial problem. Three main issues have to be addressed: (a) the operating parameters must have reasonable values (i.e. small voltages), (b) heating must be reduced as much as possible and (c) good optical access is required. After investigating various trap geometries, we found that the linear segmented trap is the most suitable for our purposes. It provides good optical access and has been used by many groups for the realization of Coulomb crystals. We investigated "the best" trap dimensions that would ensure for reasonable operating parameters stable trapping and would fulfill the planar crystal condition. From a parameter survey combined with SIMION simulations we found that the dimensions most suitable for our purposes: r0 = 3 mm and z0 = 1.5 mm.

We designed and constructed the trap electrodes and the electrical connections and fixed them inside the vacuum chamber. The chamber has 5 viewports providing optical access from all directions. The trap electrodes are made of stainless-steel rods separated by Macor parts. Each segment is independently connected to a feedthrough using a thin silver-plated copper conductor with Kapton insulation. Ultra-high vacuum conditions are realized using a turbo-molecular vacuum pump together with a rotary vane pump. We use two Littrow type ECDLs for the laser cooling (397 nm and 866 nm for repumping). Ions are loaded by laser ablation with a frequency doubled Nd:YAG pulsed laser on a 99.9% purity Ca disk target. The ions' fluorescence is recorded on a Andor iCCD camera using a custom-made Nikon microscope lens system.

Conclusion

We investigated planar Coulomb crystals for which there are very few theoretical and experimental results. Using large scale Molecular Dynamics simulations we performed an extensive study of planar Coulomb crystals in both Penning and RF traps including the effects of RF heating and collisions with background gas heating. We analyzed planar bicrystals and found out the limit imposed on the number of ions by the mass ratio and derived the dependence of spatial separation on the mass ratios and radial frequency shift

Using the results from the Molecular Dynamics simulations we estimated the error in the "push-gate" in planar Coulomb crystals. We also considered the error in quantum simulation and estimated the fidelity in the simulation of effective spin-spin interactions.

In order to realize planar Coulomb crystals in experiment, we designed and built an RF-trap system optimized for the implementation of planar crystals. The geometry of the designed linear segmented quadrupole trap provides large axial frequencies even for low voltages. Moreover, the RF-heating is drastically reduced.

本論文は,理論的に提案されている平面クーロンクリスタルを利用した量子シミュレーションを実現するための理論的考察と数値解析を中心に8章から構成されている.

第1章は序論で研究の背景と目的を述べている.量子情報処理という広い分野の中で,量子シミュレーションは必要となるキュービット数が少ないこともあり,精力的に取り組むべき課題であることを指摘している.量子コンピュータの実現可能性の高いものとして捕獲イオンを用いた手法の現状について詳細に説明されている.これらを踏まえて近年理論的に提案された捕獲イオンを用いた量子シミュレーションの手法について紹介している.これは平面クーロンクリスタルを必要とするが,それについての研究はほとんどないことから,平面クーロンクリスタルの性質およびその実現可能性を明らかにすることを本研究の目的としている.

第2章では本研究で必要となる理論について説明している.量子シミュレーションの概念について説明した後,クーロンクリスタルの定義およびその性質を述べている.イオントラップとして,静電磁場を用いたペニングトラップとrf電場を用いたrf(ポール)トラップについて,それぞれに捕獲されたイオンの挙動を運動方程式から論じている.その後,理論的に提案されている平面クーロンクリスタルを利用した量子シミュレーションの手法について説明している.これはイオンの内部状態に応じた力をそのイオンに及ぼすことにより量子ゲートを実現する手法であり,その原理の紹介と発生する誤差の評価を行っている.

第3章ではクーロンクリスタルの状態を計算するために用いた手法を説明している.まず本研究で利用した分子動力学法計算コードProtoMolについて紹介した後,ペニングトラップ,rfトラップの双方の場合について捕獲イオンに及ぼされる力および閉じ込め周波数を示している.次にイオン加熱の効果について論じている.rf電場が主要因であるが,残留ガスとの衝突の効果もモデル化することにより考慮している.既往の実験結果と比較することにより,本研究でProtoMolが正しく利用されていることを確認している.

第4章では平面クーロンクリスタルの構造と性質について論じている.イオン5,000個までを対象として,その構造を整理している.量子シミュレーションに利用できる隣同士のイオンで構成する三角形の構造と,周辺部分で顕著な大きな円を描く殻構造の二つから構成されていることを明らかにしている.またイオン数に応じた殻構造数を求めている.平面クーロンクリスタルの半径に対するイオン個数をグラフ化することで,特に外周付近で殻構造が支配的であることを示している.さらに量子シミュレーションにおいて重要なイオン間距離を平面クーロンクリスタルの半径に対して求めており,イオン個数に対する利用可能な有効半径を示している.またrf加熱によるクリスタル構造の変化を明らかにしている.

第5章では複数種類のイオンで構成される平面クーロンクリスタルの構造と性質について論じている.2種類のイオンの質量の差に対して平面クーロンクリスタルを形成できる最大イオン数を求めるとともに,その構造やイオン間距離を考察している.さらに軽いイオン(小さいm/q)からクリスタルの中心を占めていくことに基づいて異種イオン間距離を理論的に導出し,分子動力学法による数値計算と比較して良好な一致が得られている.さらに異種イオン間での熱伝達について冷却と加熱の場合の数値計算を行い,異種イオン間距離との関係を明らかにしている.

第6章では量子シミュレーションを実行する際に発生する誤差について検討を行っている.平面クーロンクリスタルで実現される閉じ込め周波数,イオン間距離,温度をパラメータとして量子シミュレーションで発生する誤差を理論的解析により定量的に明らかにしている.

第7章では平面クーロンクリスタルを実現するために必要となる電極設計を行い,その性能を数値計算により評価している.具体的には,電場解析ソフトを用いて生成ポテンシャル場を評価するとともに,その電極形状依存性を明らかにしている.さらに電源の制約なども考慮した上で電極間距離を評価している.これにより得られる閉じ込め周波数を導出し,量子シミュレーションに利用可能な範囲を明確にしている.

第7章は結論であり,本研究のまとめが述べられている.

以上を要するに,本論文は量子シミュレーションにおいて有望な平面クーロンクリスタルを用いる手法について,rfトラップにより実現する上で必要となる平面クーロンクリスタルの性質や構造について数値計算により明らかにしている.さらに電極設計を行っており,平面クーロンクリスタルおよびその量子シミュレーションへの利用に対する知見を得る上で多大な寄与をしている.

よって本論文は博士(工学)の学位請求論文として合格と認められる.