Rydberg atoms with principal quantum number n >> 1 have exaggerated atomic properties including dipole-dipole interactions that scale as n^4 and radiative lifetimes that scale as n^3. It was proposed a decade ago to take advantage of these properties to implement quantum gates between neutral atom qubits. The availability of a strong, long-range interaction that can be coherently turned on and off is an enabling resource for a wide range of quantum information tasks stretching far beyond the original gate proposal. Rydberg enabled capabilities include long-range two-qubit gates, collective encoding of multi-qubit registers, implementation of robust light-atom quantum interfaces, and the potential for simulating quantum many body physics. We review the advances of the last decade, covering both theoretical and experimental aspects of Rydberg mediated quantum information processing.Comment: accepted version, to appear in Rev. Mod. Phys., 40 figures

We present a wave function approach to study the evolution of a small system when it is coupled to a large reservoir. Fluctuations and dissipation originate in this approach from quantum jumps occurring randomly during the time evolution of the system. This approach can be applied to a wide class of relaxation operators in the Markovian regime, and it is equivalent to the standard Master Equation approach. The problem of dissipation plays a central role in Atomic Physics and Quantum Optics. The simplest example is the phenomenon of spontaneous emission, where the coupling between an atom and the ensemble of modes of the quantized electromagnetic field gives a finite lifetime to all excited atomic levels. Usually

We present a wave-function approach to the study of the evolution of a small system when it is coupled to a large reservoir. Fluctuations and dissipation originate in this approach from quantum jumps that occur randomly during the time evolution of the system. This approach can be applied to a wide class of relaxation operators in the Markovian regime, and it is equivalent to the standard master-equation approach. For systems with a number of States N much larger than unity this Monte Carlo wave-function approach can be less expensive in terms of calculation time than the master-equation treatment. Indeed, a wave function involves only N components, whereas a density matrix is described by N~ terms. We evaluate the gain in computing time that may be expected from such a formalism, and we discuss its applicability to several examples, with particular emphasis on a quantum description of laser cooling.

We propose an implementation of quantum logic gates via virtual vibrational excitations in an ion trap quantum computer. Transition paths involving unpopulated, vibrational states interfere destructively to eliminate the dependence of rates and revolution frequencies on vibrational quantum numbers. As a consequence quantum computation becomes feasible with ions whos vibrations are strongly coupled to a thermal reservoir.Pacs. 03.67.Lx, 03.65 Bz, 89.70+c Recently, methods to entangle states of two or several quantum systems in controlled ways have become subject of intense studies. Such methods may find applications in fundamental tests of quantum physics [1] and in precision spectroscopy [2]; and they offer fundamentally new possibilities in quantum communication and computing [3]. A major obstacle to these efforts is decoherence of the relevant quantum states. In many proposed implementations of quantum computation the quantum bits (qubits) are stored in physical degrees of freedom with long coherence times, like nuclear spins, and decoherence is primarily due to the environment interacting with the channel used to perform logic gates between qubits [4]. In this Letter we present a scheme which is insensitive to the interaction between the quantum channel and the environment. Specifically, we consider an implementation of quantum computation in an ion trap, but we hope to stimulate similar ideas to reduce decoherence in other physical implementations.The ion trap was originally proposed by Cirac and Zoller [5] as a system with good experimental (optical) access and control of the quantum degrees of freedom, and they suggested an implementation of the necessary ingredients in terms of one-bit and two-bit operations to carry out quantum computation. In the ion trap computer, qubits are represented by internal states of the ions. The number of qubits equals the number of ions, and this system is scalable to the problem size in contrast to NMR quantum computation which is only applicable with a limited number of qubits [6].The ion trap method [5] uses collective spatial vibrations for communication between ions, and it requires that the system is restricted to the joint motional ground state of the ions. For two ions this has recently been accomplished [7]. We present an alternative implementation of quantum gates that is both insensitive to the vibrational state and robust against changes in the vibrational motion occurring during operation, as long as the ions are in the Lamb-Dicke regime, i.e., their spatial excursions are restricted to a small fraction of the wavelength of the exciting radiation. Our mechanism relies on features of quantum mechanics that are often responsible for "paradoxical effects": i) The vibrational degrees of freedom used for communication in our scheme only enter virtually i.e., although they are crucial as intermediate states in our processes, we never transfer population to states with different vibrational excitation. ii) Transition paths involving different unpopulated, vibrationa...

We propose an efficient method to produce multi-particle entangled states of ions in an ion trap for which a wide range of interesting effects and applications have been suggested. Our preparation scheme exploits the collective vibrational motion of the ions, but it works in such a way that this motion need not be fully controlled in the experiment. The ions may, e.g., be in thermal motion and exchange mechanical energy with a surrounding heat bath without detrimental effects on the internal state preparation. Our scheme does not require access to the individual ions in the trap.