The concept of a "quantum computer" has attracted much attention in recent years. Many research groups around the world are studying the extraordinary potential of quantum computers and attempting their realisation. Current fundamental experiments are directed towards the individual building blocks of such a computer. The concept of using quantum mechanical systems to calculate complex problems was created in 1982 with Feynmans proposal of the quantum simulator. This concept represents a well controlled framework of interacting quantum systems. The intention is to map a different system of interest not mastered in the lab onto the quantum simulator. From the behaviour of the quantum simulator the behaviour of the system of interest can be deduced. A quantum mechanical measurement on the quantum simulator is thus comparable to one run of a numerical simulation of the system of interest. The difference is that a numerical simulation can often only be run under severe simplifications that then challenge the practical relevance of the result. Only very small quantum systems can be calculated on today's computers without simplifications. In this work a system of ground state atoms stored in a 3D optical lattice is presented. Each of the up to 100,000 atoms is stored in its own potential minimum, isolated from the other atoms. This state is a formidable starting point for the realisation of a quantum simulator: every atom is considered the information carrier of a spin-1/2 system. Only very few of the other concepts currently under investigation have as many information carriers as a Mott-Insulator state in an optical lattice. We have already shown in preparatory experiments a long storage time and good coherence times. Here this system is extended by a major prerequisite for a quantum simulator: controllable interactions between individual atoms are essential in order to model the interaction terms of the system of interest. These interactions are realised by a state-selective in the optical lattice by state-selective trapping potentials. If the states of an atom are called |0> and |1>, then there are two distinct potentials V0 and V1 that each act on only one of the states. The two potentials are shifted with respect to each other and thus allow bringing neighbouring atoms into contact. Since the shifting is always performed along one of the three lattice axes, the interaction is not confined to a single pair of atoms, but all pairs of neighbouring atoms along a lattice axis interact. This inherent parallelism has allowed us to create entanglement in large systems in just a single operation. The entanglement has then been measured in a Ramsey interferometer. A sequence of one nearest-neighbour interaction produces the cluster state, a maximally entangled state. More recent proposals suggest to use a cluster state as the basis for a new kind of quantum computer. As opposed to today's computers, this quantum computer would not be a Touring machine whose working algorithm is only controlled by programming. Instead the wiring of the quantum gates would have to be changed in order to solve a new problem. This is loosely comparable to today's FPGA (Field Programmable Gate Arrays - programmable logic chips). Until these quantum computers are built, quite some improvements on the experimental techniques will be necessary. But the first quantum simulators are now practically within reach.
The concept of a "quantum computer" has attracted much attention in recent years. Many research groups around the world are studying the extraordinary potential of quantum computers and attempting their realisation. Current fundamental experiments are directed towards the individual building blocks of such a computer. The concept of using quantum mechanical systems to calculate complex problems was created in 1982 with Feynmans proposal of the quantum simulator. This concept represents a well controlled framework of interacting quantum systems. The intention is to map a different system of interest not mastered in the lab onto the quantum simulator. From the behaviour of the quantum simulator the behaviour of the system of interest can be deduced. A quantum mechanical measurement on the quantum simulator is thus comparable to one run of a numerical simulation of the system of interest. The difference is that a numerical simulation can often only be run under severe simplifications that then challenge the practical relevance of the result. Only very small quantum systems can be calculated on today's computers without simplifications. In this work a system of ground state atoms stored in a 3D optical lattice is presented. Each of the up to 100,000 atoms is stored in its own potential minimum, isolated from the other atoms. This state is a formidable starting point for the realisation of a quantum simulator: every atom is considered the information carrier of a spin-1/2 system. Only very few of the other concepts currently under investigation have as many information carriers as a Mott-Insulator state in an optical lattice. We have already shown in preparatory experiments a long storage time and good coherence times. Here this system is extended by a major prerequisite for a quantum simulator: controllable interactions between individual atoms are essential in order to model the interaction terms of the system of interest. These interactions are realised by a state-selective in the optical lattice by state-selective trapping potentials. If the states of an atom are called |0> and |1>, then there are two distinct potentials V0 and V1 that each act on only one of the states. The two potentials are shifted with respect to each other and thus allow bringing neighbouring atoms into contact. Since the shifting is always performed along one of the three lattice axes, the interaction is not confined to a single pair of atoms, but all pairs of neighbouring atoms along a lattice axis interact. This inherent parallelism has allowed us to create entanglement in large systems in just a single operation. The entanglement has then been measured in a Ramsey interferometer. A sequence of one nearest-neighbour interaction produces the cluster state, a maximally entangled state. More recent proposals suggest to use a cluster state as the basis for a new kind of quantum computer. As opposed to today's computers, this quantum computer would not be a Touring machine whose working algorithm is only controlled by programming. Instead the wiring of the quantum gates would have to be changed in order to solve a new problem. This is loosely comparable to today's FPGA (Field Programmable Gate Arrays - programmable logic chips). Until these quantum computers are built, quite some improvements on the experimental techniques will be necessary. But the first quantum simulators are now practically within reach.
Atoms in optical lattices are promising candidates to implement quantum information processing. Their behaviour is well understood on a microscopic level, they exhibit excellent coherence properties, and they can be easily manipulated using external fields. In very deep optical lattices, each atom is restricted to a single lattice site and can be used as a qubit. If the lattice is shallow enough such that the atoms can move, their properties can be used to simulate certain condensed matter phenomena such as superconductivity. In this thesis, we show how technical problems of optical lattices such as restricted decoherence times, or fundamental shortcomings such as the lack of phonons or strong spin interactions, can be overcome by using current or near-future experimental techniques. We introduce a scheme that makes it possible to simulate model Hamiltonians known from high-temperature superconductivity. For this purpose, previous simulation schemes to realise the spin interaction terms are extended. We especially overcome the condition of a filling factor of exactly one, which otherwise would restrict the phase of the simulated system to a Mott-insulator. This scheme makes a large range of parameters accessible, which is difficult to cover with a condensed matter setup. We also investigate the properties of optical lattices submerged into a Bose-Einstein condensate (BEC). A weak-coupling expansion in the BEC-impurity interaction strength is used to derive a model that describes the lattice atoms in terms of polarons, i.e.~atoms dressed by Bogoliubov phonons. This is analogous to the description of electrons in solids, and we observe similar effects such as a crossover from coherent to incoherent transport for increasing temperatures. Moreover, the condensate mediates an attractive off-site interaction, which leads to macroscopic clusters at experimentally realistic parameters. Since the atoms in the lattice can also be used as a quantum register with the BEC mediating a two-qubit gate, we derive a quantum master equation to examine the coherence properties of the atomic qubits. We show that the system exhibits sub- and superdecoherence and that a fast implementation of the two-qubit gate competes with dephasing. Finally, we show how to realise the encoding of qubits in a decoherence-free subspace (DFS) using optical lattices. We develop methods for implementing robust gate operations on qubits encoded in a DFS exploiting collisional interactions between the atoms. We also give a detailed analysis of the performance and stability of the gate operations and show that a robust implementation of quantum repeaters can be achieved using our setup. We compare the robust repeater scheme to one that makes use of conventional qubits only, and show the conditions under which one outperforms the other.
Quantum many-body systems, as they appear in different branches of physics and chemistry, are very hard to simulate. The number of parameters defining their quantum states grows exponentially with the number of constituents. This prevents us from exploring and investigating relevant problems involving relatively small numbers of subsystems even with the help of the most powerful supercomputers. Richard Feynman already noticed this difficulty back in 1981 when he wrote his visionary paper “Simulating Physics with Computers” [Int. J. Theor. Phys. 21:6/7, 467 (1982)]. There, he proposed to use quantum systems (as opposed to classical computers) to perform the simulation and in this way to circumvent the need to store and compute a colossal number of parameters, necessary to describe the pertaining superpositions. What was a visionary idea at that time has become a very active field of research in the last few years. The extraordinary progress of both theoretical and experimental research in quantum physics allows us now to tame, control, and manipulate various quantum systems with unprecedented precision. Laser, magnetic, electronic and other technologies enable us to arrange quantum subsystems in different geometries, modify their interactions, and detect them. This allows us to engineer Hamiltonians and emulate systems displaying some of the most intriguing quantum phenomena. This tremendous progress is, to a large extent, due to the pioneering work of many researchers during the last twenty to thirty years. Here, the investigations of basic phenomena with one or few atoms or photons were of particular impact. The degree of control achieved in the experiments has enabled the observation of mind-boggling quantum effects, related to the superposition principle or to entangled states, and their disappearance due to decoherence. In fact, the 2012 Nobel Prize has been awarded to two scientists, S. Haroche and D. Wineland, for such ground-breaking experiments. Such experiments, often in combination with laser cooling of atoms or ions and Bose-Einstein condensation, have formed one of the pillars on which the field of quantum simulations is being established. Quantum simulations with cold atoms, either in magnetic traps or in optical lattices, with trapped ions, with photons, with superconducting devices, with quantum dots, etc., have been theoretically proposed and experimentally considered by many researchers. Most of the proposals so far deal with problems in condensed matter physics that are either difficult to tackle with modern supercomputers or are difficult to observe in solid state systems. This has established a very close link between different disciplines, like atomic, molecular, and optical (AMO) physics, and condensed matter physics. Although some of the experiments are still in their infancy, and a lot of research is still needed, the extraordinary progress made in all these research areas during the last few years makes us feel confident that in the not too distant future quantum simulators will provide a key tool to investigate many-body quantum systems. Furthermore, we also expect that quantum simulators will help quantum physicists to establish new and fruitful links to other scientific communities, like high-energy physics or quantum chemistry. The present special issue contains various review papers as well as research papers on quantum simulations. It starts out with two papers written by last year's Nobel Prize winners describing their foundational work (Wineland, p. 739 and Haroche, p. 763). The issue continues with three thorough, invited review papers covering different topics on quantum simulations. The first one reviews recent theoretical proposals to use cold atoms in optical lattices to simulate lattice gauge theories of the sort that appear in high-energy physics (U.-J. Wiese, p. 783). The second covers both theoretical proposals and experimental demonstrations of cold-atom systems to simulate the physics of matter in the presence of gauge fields, displaying so-called topological phenomena, which so far have only been observed in solid-state systems (I. Spielman, p. 794). The third one contains an extensive overview and new proposals to perform a broad range of quantum simulations using quantum dot systems (P. Barthelmy et al., p. 813). The issue also includes six original papers with a variety of theoretical proposals as well as the introduction of novel theoretical techniques related to the field of quantum simulation. The field of quantum simulations is just starting, but we are confident that it will soon generate an even larger plethora of new ideas and discoveries, trigger interdisciplinary research, and lead us to the adventure of describing and understanding the quantum world. We hope that this special issue will help future researchers in such an exciting adventure. Rainer Blatt received his PhD degree in physics from the University of Mainz in 1981. As a postdoctoral fellow he worked on atomic beam cooling in Boulder, CO and later as a research assistant in Hamburg with single trapped ions. In 1995 he accepted a chair in experimental physics at the University of Innsbruck working on quantum information processing with trapped ions. Since 2003 he is also scientific director at the Institute for Quantum Optics and Quantum Information (IQOQI) of the Austrian Academy of Sciences. Immanuel Bloch received his PhD degree in physics from the Ludwig Maximilians University of Munich in 2000. As a junior group leader, he continued in Munich starting his work on ultracold quantum gases in optical lattices. In 2008 he was appointed scientific director of the newly founded division on Quantum Many-Body Systems at the Max Planck Institute of Quantum Optics. Since 2009 he also holds a chair in quantum optics at the Ludwig-Maximilians University of Munich. Ignacio Cirac received his PhD degree in physics from the Universidad Complutense de Madrid in 1991. He was professor at the Universities of Castilla-La Mancha and Innsbruck from 1991–1996 and 1996–2001, respectively. In 2001 he joined the Max-Planck Institute of Quantum Optics, where he leads the Theory Division. His current research interests include quantum information theory and quantum optics. Peter Zoller obtained his doctorate from the University of Innsbruck in 1977. He was Max Kade Fellow at the University of Southern California from 1978/79, and was JILA visiting Fellow 1981/82 and 1988. In 1990 he became JILA Fellow and professor of physics at the University of Colorado. At the end of 1994, he accepted a chair at the University of Innsbruck, where he has worked ever since. Since 2003, he has also held the position of Scientific Director at the Institute for Quantum Optics and Quantum Information (IQOQI) of the Austrian Academy of Sciences.
Many outstanding problems in quantum physics, such as high-Tc superconductivity or quark confinement, are still - after decades of research - awaiting commonly accepted explanations. One reason is that such systems are often difficult to control, show an intermingling of several effects, or are not easily accessible to measurement. To arrive at a deeper understanding of the physics at work, researchers typically derive simplified models designed to capture the most striking phenomena of the system under consideration. However, due to the exponential complexity of Hilbert space, even some of the simplest of such models pose formidable challenges to analytical and numerical calculations. In 1982, Feynman proposed to solve such quantum models with experimental simulation on a physically distinct, specifically engineered quantum system [Int. J. Theor.Phys. 21, 467]. Designed to be governed by the same underlying equations as the original model, it is hoped that direct measurements on these so called quantum simulators (QSs) will allow to gather deep insights into outstanding problems of physics and beyond. In this thesis, we identify four requirements that a useful QS has to fulfill, relevance, control, reliability, and efficiency. Focusing on these, we review the state of the art of two popular approaches, digital QSs (i.e., special purpose quantum computers) and analog QSs (devices with always-on interactions). Further, focusing on possibilities to increase control over QSs, we discuss a scheme to engineer quantum correlations between mesoscopic numbers of spinful particles in optical lattices. This technique, based on quantum polarization spectroscopy, may be useful for state preparation and quantum information protocols. Additionally, employing several analytical and numerical methods for the calculation of many-body ground states, we demonstrate the variety of condensed-matter problems that can be attacked with QSs consisting of ultracold ions or neutral atoms in optical lattices. The chosen examples, some of which have already been realized in experiment, include such diverse settings as frustrated antiferromagnetism, quantum phase transitions in exotic lattice geometries, topological insulators, non-Abelian gauge-fields, orbital order of ultracold Fermions, and systems with long-range interactions. The experimental realization of all of these models requires techniques which go beyond standard optical lattices, e.g., time-periodic driving of lattices with exotic geometry, loading ultracold atoms into higher bands, or immersing trapped ions into an optical lattice. The chosen models, motivated by important open questions of quantum physics, pose difficult problems for classical computers, but they may be amenable in the near future to quantum simulation with ultracold atoms or ions. While the experimental control over relevant models has increased dramatically in the last years, the reliability and efficiency of QSs has received considerably less attention. As a second important part of this thesis, we emphasize the need to consider these aspects under realistic experimental conditions. We discuss specific situations where terms that have typically been neglected in the description of the QS introduce systematic errors and even lead to novel physics. Further, we characterize in a generic example the influence of quenched disorder on an analog QS. Its performance for simulating universal behavior near a quantum phase transition seems satisfactory for low disorder. Moreover, our results suggest a connection between the reliability and efficiency of a QS: it works less reliable exactly in those interesting regimes where classical calculations are less efficient. If QSs fulfill all of our four requirements, they may revolutionize our approach to quantum-mechanical problems, allowing to solve the behavior of complex Hamiltonians, and to design nano-scale materials and chemical compounds from the ground up.
In this chapter, we present a proposal for the realization of an entangling gate for photons. The protocol works deterministically and is experimentally feasible under realistic conditions. We start outlining the general problem of quantum information processing with photons and describe the current status in this field. Then we introduce the proposed scheme and present its main features. This part is followed by a review of the concepts and procedures employed in our proposal. Finally, we explain the quantum gate, in detail, in the last section of this chapter, where we also briefly discuss the performance of the gate in the presence of imperfections. 26.1.1 Quantum information processing with light Photons play a key role in applications of quantum information because they are ideally suited to transmit quantum states between distant sites. This feature makes them indispensable for the realization of quantum communication protocols and the construction of quantum networks. While being a good flying carrier of information, photons are naturally less adequate for storage than the long-lived matter degrees of freedom. For this reason, matter systems are employed as quantum memories. In order to combine both elements in a quantum network (i.e., photons as flying qubits and atoms as memory devices), light-matter interface schemes have been developed to transfer quantum states of light onto an atomic system. Some of these schemes are based on quantum-nondemolition interactions, electromagnetically induced transparency, and Raman processes. Moreover, Raman processes have been used to entangle two distant atomic ensembles, which is an important step to realize quantum repeaters and, thus to solve the problem of losses and decoherence that exists in a photonic channel. Apart from the transmission and storage of information, in a quantum network it is necessary to process quantum states. However, manipulation of quantum states of light is still challenging, since it requires the ability to create entanglement between photons. This task is difficult because photons are noninteracting particles, in principle. One possibility to entangle photons is to employ materials that possess optical nonlinearities, but so far, there are no materials available whose nonlinearities are strong enough to allow for short gate times. An alternative approach was put forward by Knill et al., which requires only linear optical operations and measurements. However, this scheme is probabilistic and not very efficient in practice. In this chapter, we present a scheme for the realization of a deterministic entangling gate for photons by using an atom lattice. An atom lattice consists of an ensemble of cold atoms loaded in the periodic optical potential created by a standing wave. In order to use the lattice as a quantum register for quantum computation, it is necessary to prepare it in a Mott insulating phase, in which the number of atoms in each site of the potential (i.e., in each of its minima) is approximately constant, and set equal to 1.
We review our experiments on quantum information processing with neutral atoms in optical lattices and magnetic microtraps. Atoms in an optical lattice in the Mott insulator regime serve as a large qubit register. A spin-dependent lattice is used to split and delocalize the atomic wave functions in a controlled and coherent way over a defined number of lattice sites. This is used to experimentally demonstrate a massively parallel quantum gate array, which allows the creation of a highly entangled many-body cluster state through coherent collisions between atoms on neighbouring lattice sites. In magnetic microtraps on an atom chip, we demonstrate coherent manipulation of atomic qubit states and measure coherence lifetimes exceeding one second at micron-distance from the chip surface. We show that microwave near-fields on the chip can be used to create state-dependent potentials for the implementation of a quantum controlled phase gate with these robust qubit states. For single atom detection and preparation, we have developed high finesse fiber Fabry-Perot cavities and integrated them on the atom chip. We present an experiment in which we detected a very small number of cold atoms magnetically trapped in the cavity using the atom chip.
A quantum simulator is a well-controlled quantum system which can be used to simulate another quantum system. Quantum degenerate gases confined by optical standing waves(optical lattices) have been considered to simulate condensed matter systems. We focus on two issues in this problem. The first one is the realization of interesting quantum phases by engineering Hamiltonians with the cold atoms toolbox. Secondly, the systems we studied are often intractable to analytical methods, which makes it a crucial issue to solve them with numerical methods. For the first issue we find several interesting phases can be studied with cold atoms in optical lattices. We show that a p-wave superfluid is stablized through a dissipation blockade mechanism. This mechanism also induces a new insulator formed by p-wave Feshbach molecules. We find that the anisotropic nature of interaction between polar molecules or dipolar atoms can be used to realize two charge density waves with different patterns and a supersolid phase which has not been found conclusively in the condensed matter system. Finally, we propose to realize a long sought spin liquid phase with hard-core bosons subject to a spin-dependent lattice and Raman induced hoppings. For the second issue we choose to numerically study these systems with the so-called tensor network algorithm which is a variational method based on the matrix(tensor) product states product states which are designed to approximate generic wave functions from their entanglement properties. In general the tensor network algorithm has advantages that it is applicable beyond one dimension, and can be used to study physical properties in the thermodynamic limit directly. We apply this method to map out the phase diagrams of the above systems. Furthermore, we show that it is capable of capturing the phase diagram of a frustrated magnetism model, the Heisenberg model on a checkerboard lattice. The great controllability and isolation from the environment of a cold atom system offer a playground to study interesting physics in strongly correlated systems. Several proposed control schemes in this thesis have been demonstrated in experiments. By providing detailed evidence from numerical simulations, this work addresses practical routes to novel quantum phases.