In optically controlled quantum computers it may be favorable to address different qubits using light with different frequencies, since the optical diffraction does not then limit the distance between qubits. Using qubits that are close to each other enables qubit-qubit interactions and gate operations that are strong and fast in comparison to qubit-environment interactions and decoherence rates. However, as qubits are addressed in frequency space, great care has to be taken when designing the laser pulses, so that they perform the desired operation on one qubit, without affecting other qubits. Complex hyperbolic secant pulses have theoretically been shown to be excellent for such frequency-addressed quantum computing [I. Roos and K. Molmer, Phys. Rev. A 69, 022321 (2004)]---e.g., for use in quantum computers based on optical interactions in rare-earth-metal-ion-doped crystals. The optical transition lines of the rare-earth-metal-ions are inhomogeneously broadened and therefore the frequency of the excitation pulses can be used to selectively address qubit ions that are spatially separated by a distance much less than a wavelength. Here, frequency-selective transfer of qubit ions between qubit states using complex hyperbolic secant pulses is experimentally demonstrated. Transfer efficiencies better than 90% were obtained. Using the complex hyperbolic secant pulses it was also possible to create two groups of ions, absorbing at specific frequencies, where 85% of the ions at one of the frequencies was shifted out of resonance with the field when ions in the other frequency group were excited. This procedure of selecting interacting ions, called qubit distillation, was carried out in preparation for two-qubit gate operations in the rare-earth-metal-ion-doped crystals. The techniques for frequency-selective state-to-state transfer developed here may be also useful also for other quantum optics and quantum information experiments in these long-coherence-time solid-state systems.
In optically controlled quantum computers it may be favorable to address different qubits using light with different frequencies, since the optical diffraction does not then limit the distance between qubits. Using qubits that are close to each other enables qubit-qubit interactions and gate operations that are strong and fast in comparison to qubit-environment interactions and decoherence rates. However, as qubits are addressed in frequency space, great care has to be taken when designing the laser pulses, so that they perform the desired operation on one qubit, without affecting other qubits. Complex hyperbolic secant pulses have theoretically been shown to be excellent for such frequency-addressed quantum computing [I. Roos and K. Molmer, Phys. Rev. A 69, 022321 (2004)]---e.g., for use in quantum computers based on optical interactions in rare-earth-metal-ion-doped crystals. The optical transition lines of the rare-earth-metal-ions are inhomogeneously broadened and therefore the frequency of the excitation pulses can be used to selectively address qubit ions that are spatially separated by a distance much less than a wavelength. Here, frequency-selective transfer of qubit ions between qubit states using complex hyperbolic secant pulses is experimentally demonstrated. Transfer efficiencies better than 90% were obtained. Using the complex hyperbolic secant pulses it was also possible to create two groups of ions, absorbing at specific frequencies, where 85% of the ions at one of the frequencies was shifted out of resonance with the field when ions in the other frequency group were excited. This procedure of selecting interacting ions, called qubit distillation, was carried out in preparation for two-qubit gate operations in the rare-earth-metal-ion-doped crystals. The techniques for frequency-selective state-to-state transfer developed here may be also useful also for other quantum optics and quantum information experiments in these long-coherence-time solid-state systems.
The main aim of this work, was to lay the foundations for the experimental realisation of a quantum mechanical controlled NOT gate in rare-earth-metal-ion-doped crystals. Small amounts of rare-earth elements, added during the growth of some inorganic crystals, will become substituted into the crystal lattice as trivalent ions. The trivalent rare-earth-metal ions between cerium, with atomic number 58, and ytterbium, with atomic number 70, have a partly filled 4f shell, which does not extend spatially outside the full 5s and 5p shells. The 4f vacancies make electronic inner shell transitions possible between spectroscopic 4f terms. Some of these optical transitions have coherence times of the order of milliseconds, when the crystals are cooled down to ~ 4 K. There are several reasons for these extraordinary coherence times, which are approximately 8 orders of magnitude greater than those typical for electronic transitions in solids. The most important one is the cage-like shield which the outer 5s and 5p shells provide for the 4f electrons. Furthermore, since these ions are naturally trapped inside the crystal lattice there is no Doppler broadening of the line-width. The coherence properties of these optical transitions is one of the features that makes these materials attractive for use as a solid-state platform for quantum computing, using these ions as qubits. Another appealing characteristic is the fact that different ions have different optical resonance frequencies, which means that ions belonging to different qubits, which only have nm separation, can still be addressed separately by using different laser frequencies. Since the inter-ion spacing is so small, it is possible to make two ions interact strongly, although they are well shielded, through a permanent dipole-dipole interaction. This interaction can be turned on and off by switching between two different ways of encoding the qubit, a most useful feature. When the qubit is represented as a superposition between two ground state hyperfine levels, the interaction is turned off. The interaction is turned on selectively by transferring this superposition to the optical transition with a pi-pulse, for the specific ions that are to interact. This thesis describes how peaks of ions, absorbing on a single transition, residing in spectral pits with no other ions, have been isolated. It is shown how these ions can be coherently transferred between hyperfine levels via the optically excited state, how the interaction between such peaks of ions representing qubits can be turned on and off, and how subgroups of ions with strong interaction can be distilled out. All the work described here has been performed using the ensemble approach. The ensemble approach will, however, be difficult to scale up to large numbers of qubits. A method employing a single ion in each qubit, using a specialised ion for readout, has therefore also been proposed. The rare-earth-metal-ion-based quantum computing experiments require a laser with coherence properties which at least match those of the material. To this end a frequency stabilisation system was developed for a dye laser. This system uses a transient spectral hole in a rare-earth-metal-ion-doped crystal, of the same kind that is used in the experiments, as frequency reference, and is to the authors knowledge the first demonstration of locking a dye laser to a spectral hole. This system provides a line-width of 1 kHz on a 10 microseconds timescale and a frequency drift below 1 kHz/s.
The proximity between the ions results in large interactions and potentially allows fast gates, but they can still be separately addressed since different ions have different optical resonance frequency. The interaction that enables multi-qubit gates can be turned on at will and is based on that the permanent dipole moment changes as a control ion is transferred to the optically excited state which in turn Stark-shifts target qubit out of resonance. Using optical pumping, all ions within a frequency interval can be removed and then a peak of equivalent ions, each belonging to one instance of many parallel quantum computers can be positioned within the non-absorbing region. This qubit has then been efficiently transferred between different qubit states using robust complex hyperbolic secant pulses. Pairs of qubits that interact strongly have also been distilled
This thesis describes investigations of the use of cryogenically cooled rare-earth-ion-doped crystals for quantum information processing and quantum optics. Several aspects of the coherent interaction between light and rare-earth ions in solids are addressed. Quantum information science has given physicists new views of quantum mechanics. The transmission of quantum states has already found practical use and full scale quantum computers may one day perform computations and simulations that would be impossible on a conventional computer. The work presented in this thesis can be seen as a part of a broad effort to learn how to control and manipulate quantum mechanical systems, which will become necessary as science and technology continue to push ever deeper into the nanoscopic world. Coherent radiation, such as laser light, provides us with an ideal tool for these investigations and, along the way, we may also learn more about the quantum nature of light. Rare-earth ions in inorganic crystals have several unusual properties that are interesting for applications within quantum information science, including long coherence times and long-lived ground state sublevels that can be used for storage of quantum and classical information. As part of the work presented in this thesis, new materials have been investigated with respect to these properties, and ways to enhance the useful properties of the materials were explored. In one investigation, the lifetime of information stored in the ground state population distribution of Tm3 ions in YAG was shown to increase by several orders of magnitude with the application of a magnetic field. It is demonstrated how the optical inhomogeneous absorption profile can be prepared, so that the light only interacts with a selected group of ions, absorbing on a specific transition. Narrow absorbing structures, with widths approaching the optical homogeneous linewidth, have been prepared with no absorption in the surrounding spectral interval. This thesis addresses the use of such structures as hardware for quantum bits. Tailored pulses, capable of inducing controlled changes in the quantum states of the ions (qubits), even in the presence of unknown variations of coupling strengths and frequencies, have been realised experimentally and used for multiple transfer of ions between energy levels. Ion-ion interactions, which can be used for performing quantum logic operations, have been investigated in some detail. Techniques for selecting strongly interacting ions, by transferring weakly interacting ions to auxiliary states, have been demonstrated. A scheme for storing the quantum state of light in a solid, using photon-echo-like techniques, is proposed and analysed. In the proposed scheme, an optical wave packet is absorbed and subsequently re-emitted by an inhomogeneous absorption profile, which is tailored and externally controlled by the application of an electric field Additionally, an accumulated photon echo experiment has been performed using faint optical pulses. The experiment can be viewed as a demonstration of delayed self-interference of a single photon and as a demonstration of how a single photon can act as two of the fields in a photon echo process.
We present a scheme for quantum computing in rare-earth-ion doped inorganic crystals. We use two of the ground state hyperfine levels as the qubit states, with different qubits absorbing light at different frequencies within an inhomogeneously broadened optical transition. Interaction between different qubits is accomplished using the change in permanent dipole moment that is induced at the optical excitation of the ions. The two most important aspects of the scheme have been accomplished experimentally: The isolation of a single qubit in a pure state and interaction between ions absorbing at different frequencies.
In this thesis, crystals of yttrium orthosilicate (Y2SiO5) that are randomly doped with another rare-earth element, such as praseodymium (Pr), europium (Eu), or cerium (Ce), are investigated with lasers locked to ultra-stable cavities using the Pound-Drever-Hall locking technique. Many of these rare-earth elements have long-lived 4f-4f transitions, hundreds of microseconds to a few milliseconds, with even longer ground hyperfine lifetimes of up to several days. The coherence properties are, to the best of the author's knowledge, the longest achieved for any material, currently with a record of six hours for the nuclear spin states of Eu3+:Y2SiO5 measured at cryogenic temperatures. Furthermore, due to natural trapping and differences in the local environments, each dopant ion experiences a slightly different crystal field, and thus an inhomogeneity in the 4f-4f transition exists between all ions in a crystal. Since the homogeneous linewidth is in the order of kHz or below, whereas the inhomogeneous profile can be several GHz wide, these materials have dense storing capabilities.This thesis explores how light interacts with such rare-earth-ion-doped crystals; how the absorption and light polarization varies during propagation; how spectral features in the inhomogeneous absorption profile can be tailored to create narrowband spectral filters; how the speed of light is slowed down significantly in such narrow transmission windows; and how that can be used to either frequency shift incoming light, control its group velocity, or temporally compress pulses. It also uses rare-earth-ions to research quantum computing, the field of using quantum mechanical effects such as superpositions and entanglements to outperform classical computers on certain specific problems. This is done by examining how two-color pulses can be used to rapidly induce coherence from an initially mixed state; how qubit-qubit interactions can be performed experimentally using ensemble qubits, which opens the door to two-qubit experiments such as the CNOT-gate and entanglements; how a scalable quantum computer might be constructed using a single ion qubit approach with a dedicated readout ion and buffer ion(s) to improve readout fidelity; and how cerium is investigated as a candidate for such a dedicated readout ion.
When doped into solid state transparent crystals, rare-earth ions can have optically excited states with milliseconds coherence time and up to six hours coherence time for hyperfine levels after being cooled down to liquid helium temperature. This long coherence property makes them unique as solid state material and attractive for quantum information applications.The rare-earth ions have narrow homogeneous linewidths on the order of ~ kHz. When doped into a crystal, the lattice structure is slightly distorted due to the size mismatch between the dopant and the host ions, resulting in slight differences in the local crystal fields at the dopant sites. Therefore, the resonance frequencies of the rare-earth ions are shifted relatively, resulting in an inhomogeneously broadened absorption profile, typically on the order of a few GHz. This provides the possibility of addressing each group of ions individually and engineering the absorption profile for different purposes by optical pumping using a narrow linewidth laser, and makes rare-earth-ion-doped crystals suitable for ensemble based quantum memories.Quantum memories are vital components for developing longdistance quantum communication. Part of this thesis project focused on making quantum memories with high efficiency and long storage time, especially using materials with low optical depth. To achieve this goal, a low finesse, unsymmetrical cavity was employed and experiments were carried out at (or close to) the impedance matched condition. Atomic frequency comb (AFC) protocol proposed for such an inhomogeneously broadened system was used to store coherent light for a certain time. 56% overall efficiency was achieved, and the main limiting factor for reaching even higher efficiency was the inefficient coupling between incoming light and the cavity mode. Other possible loss mechanisms were analyzed as well.Long-time, on-demand, quantum memory experiments were conducted with the help of two efficient radio-frequency rephasing pulses. Storage time was extended from microseconds to milliseconds after compensating (most of) the residual earth magnetic field in the lab. The opportunities and challenges of combing the cavity assistant quantum memory and the long time, on demand quantum memory were analyzed and possible solutions were proposed. Another part of the thesis project involves slow light-based applications in rare-earth-ion-doped crystals. The strong slow light effect at the presence a spectral hole is due to the fact that the light interacts with the absorbing ions outside the spectral hole off-resonantly, and most of the light energy is stored inside the ions as a polarization. For example, with a group velocity of 300 km/s, 99.9% of the pulse energy is stored in the ions when the pulse propagates inside the crystal. The slower the light propagates, the more energy is stored inside the medium. Based on this, a frequency shifter that is capable of shifting the light frequency of ± 4 MHz was demonstrated, and frequency shift up to GHz was proposed. Group velocity control by a factor of 20 and pulse compression in time by a factor of 2 were also demonstrated. The frequency shift and group velocity of the light are solely controlled by the external electric field, which makes them ideal for weak light situation. The unique property of such a frequency shifter and group velocity controller is that they could be made to accept light from an arbitrary incoming angle and polarization, for example, scattered light.