Quantum mechanics allows for many-particle wavefunctions that cannot be factorized into a product of single-particle wavefunctions, even when the constituent particles are entirely distinct. Such 'entangled' states explicitly demonstrate the non-local character of quantum theory, having potential applications in high-precision spectroscopy, quantum communication, cryptography and computation. In general, the more particles that can be entangled, the more clearly nonclassical effects are exhibited--and the more useful the states are for quantum applications. Here we implement a recently proposed entanglement technique to generate entangled states of two and four trapped ions. Coupling between the ions is provided through their collective motional degrees of freedom, but actual motional excitation is minimized. Entanglement is achieved using a single laser pulse, and the method can in principle be applied to any number of ions.

Local realism is the idea that objects have definite properties whether or not they are measured, and that measurements of these properties are not affected by events taking place sufficiently far away. Einstein, Podolsky and Rosen used these reasonable assumptions to conclude that quantum mechanics is incomplete. Starting in 1965, Bell and others constructed mathematical inequalities whereby experimental tests could distinguish between quantum mechanics and local realistic theories. Many experiments have since been done that are consistent with quantum mechanics and inconsistent with local realism. But these conclusions remain the subject of considerable interest and debate, and experiments are still being refined to overcome 'loopholes' that might allow a local realistic interpretation. Here we have measured correlations in the classical properties of massive entangled particles (9Be+ ions): these correlations violate a form of Bell's inequality. Our measured value of the appropriate Bell's 'signal' is 2.25 +/- 0.03, whereas a value of 2 is the maximum allowed by local realistic theories of nature. In contrast to previous measurements with massive particles, this violation of Bell's inequality was obtained by use of a complete set of measurements. Moreover, the high detection efficiency of our apparatus eliminates the so-called 'detection' loophole.

We have investigated motional heating of laser-cooled 9 Be + ions held in radio-frequency (Paul) traps. We have measured heating rates in a variety of traps with different geometries, electrode materials, and characteristic sizes.The results show that heating is due to electric-field noise from the trap electrodes which exerts a stochastic fluctuating force on the ion. The scaling of the heating rate with trap size is much stronger than that expected from a spatially uniform noise source on the electrodes (such as Johnson noise from external circuits), indicating that a microscopic uncorrelated noise source on the electrodes (such as fluctuating patch-potential fields) is a more likely candidate for the source of heating.

We demonstrate a decoherence-free quantum memory of one qubit. By encoding the qubit into the decoherence-free subspace (DFS) of a pair of trapped 9Be+ ions, we protect the qubit from environment-induced dephasing that limits the storage time of a qubit composed of a single ion. We measured the storage time under ambient conditions and under interaction with an engineered noisy environment and observed that encoding into the DFS increases the storage time by up to an order of magnitude. The encoding reversibly transfers an arbitrary qubit stored in a single ion to the DFS of two ions.

We experimentally investigate three methods, utilizing different atomic observables and entangled states, to increase the sensitivity of rotation angle measurements beyond the "standard quantum limit" for nonentangled states. All methods use a form of quantum mechanical "squeezing." In a system of two entangled trapped (9)Be(+) ions we observe a reduction in uncertainty of rotation angle below the standard quantum limit for all three methods including all sources of noise. As an application, we demonstrate an increase in precision of frequency measurement in a Ramsey spectroscopy experiment.

We have investigated ion dynamics associated with a dual linear ion trap where ions can be stored in and moved between two distinct locations. Such a trap is a building block for a system to engineer arbitrary quantum states of ion ensembles. Specifically, this trap is the unit cell in a strategy for scalable quantum computing using a series of interconnected ion traps. We have transferred an ion between trap locations 1.2 mm apart in 50 $\mu$s with near unit efficiency ($> 10^{6}$ consecutive transfers) and negligible motional heating, while maintaining internal-state coherence. In addition, we have separated two ions held in a common trap into two distinct traps.

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We show how an experimentally realized set of operations on a single trapped ion is sufficient to simulate a wide class of Hamiltonians of a spin-1/2 particle in an external potential. This system is also able to simulate other physical dynamics. As a demonstration, we simulate the action of an n-th order nonlinear optical beamsplitter. Two of these beamsplitters can be used to construct an interferometer sensitive to phase shifts in one of the interferometer beam paths. The sensitivity in determining these phase shifts increases linearly with n, and the simulation demonstrates that the use of nonlinear beamsplitters (n=2,3) enhances this sensitivity compared to the standard quantum limit imposed by a linear beamsplitter (n=1).One of the motivations behind Feynman's proposal for a quantum computer [1] was the possibility that one quantum system could efficiently simulate the behavior of other quantum systems. This idea was verified by Lloyd [2] and further explored by Lloyd and Braunstein [3] for a conjugate pair of variables such as position and momentum of a quantum particle. Following this suggestion we show below that coherent manipulation of the quantized motional and internal states of a single trapped ion using laser pulses can simulate the more general quantum dynamics of a single spin-1/2 particle in an arbitrary external potential. Previously, harmonic and anharmonic oscillators have been simulated in NMR [4].In addition to demonstrating the basic building blocks for simulating such arbitrary dynamics, we experimentally simulated the action of optical Mach-Zehnder interferometers with linear and nonlinear second-and thirdorder beam-splitters on number-states. Interferometers with linear beamsplitters and nonclassical input states have engendered considerable interest, since their noise limits for phase estimation can lie below the standard quantum limit for linear interferometers with coherent input modes [5][6][7][8] as has been demonstrated in experiments [9]. A number of optics experiments have exploited the second-order process of spontaneous parametric downconversion [10], which can be regarded as a nonlinear beamsplitter. By cascading this process, a fourth-order interaction has also recently been realized [11]. One difficulty in these experiments is the exponential decrease in efficiency as the order increases, necessitating data postselection and long integration times. In the simulations reported here, nonlinear interactions were implemented with high efficiency, eliminating the need for data postselection and thereby requiring relatively short integration times.To realize a quantum computer for simulating a spin s = 1/2 particle of mass µ in an arbitrary potential, one must be able to prepare an arbitrary input statewhere the particle's position wavefunction is expanded in energy eigenstates |n of a suitable harmonic oscillator and |m s (m s ∈ {↓, ↑}) represent the spin eigenstates in a suitable basis. We have recently demonstrated a method to generate arbitrary states of the type in Eq. (1)...