In order to provide a general framework within which the dispersal of cells or organisms can be studied, we introduce two stochastic processes that model the major modes of dispersal that are observed in nature. In the first type of movement, which we call the position jump or kangaroo process, the process comprises a sequence of alternating pauses and jumps. The duration of a pause is governed by a waiting time distribution, and the direction and distance traveled during a jump is fixed by the kernel of an integral operator that governs the spatial redistribution. Under certain assumptions concerning the existence of limits as the mean step size goes to zero and the frequency of stepping goes to infinity the process is governed by a diffusion equation, but other partial differential equations may result under different assumptions. The second major type of movement leads to what we call a velocity jump process. In this case the motion consists of a sequence of "runs" separated by reorientations, during which a new velocity is chosen. We show that under certain assumptions this process leads to a damped wave equation called the telegrapher's equation. We derive explicit expressions for the mean squared displacement and other experimentally observable quantities. Several generalizations, including the incorporation of a resting time between movements, are also studied. The available data on the motion of cells and other organisms is reviewed, and it is shown how the analysis of such data within the framework provided here can be carried out.

The quantum walk is the quantum analogue of the well-known random walk, which forms the basis for models and applications in many realms of science. Its properties are markedly different from the classical counterpart and might lead to extensive applications in quantum information science. In our experiment, we implemented a quantum walk on the line with single neutral atoms by deterministically delocalizing them over the sites of a one-dimensional spin-dependent optical lattice. With the use of site-resolved fluorescence imaging, the final wave function is characterized by local quantum state tomography, and its spatial coherence is demonstrated. Our system allows the observation of the quantum-to-classical transition and paves the way for applications, such as quantum cellular automata.Interference phenomena with microscopic particles are a direct consequence of their quantum-mechanical wave nature [1,2,3,4,5]. The prospect to fully control quantum properties of atomic systems has stimulated ideas to engineer quantum states that would be useful for applications in quantum information processing, for example, and also would elucidate fundamental questions, such as the quantum-to-classical-transition [6]. A prominent example of state engineering by controlled multipath interference is the quantum walk of a particle [7]. Its classical counterpart, the random walk, is relevant in many aspects of our life providing insight into diverse fields: It forms the basis for algorithms [8], describes diffusion processes in physics or biology [8,9], such as Brownian motion, or has been used as a model for stock market prices [10]. Similarly, the quantum walk is expected to have implications for various fields, for instance, as a primitive for universal quantum computing [11], systematic quantum algorithm engineering [12] or for deepening our understanding of the efficient energy transfer in biomolecules for photosynthesis [13].Quantum walks have been proposed to be observable in several physical systems [12,14,15]. Special realizations have been reported in either the populations of nuclear magnetic resonance samples [16,17]; or in optical systems, in either frequency space of a linear optical resonator [18], with beam splitters [19], or in the continuous tunneling of light fields through waveguide lattices [20]. Recently, a three-step quantum walk in the phase space of trapped ions has been observed [21]. However, the coherent walk of an individual quantum particle with controllable internal states as originally proposed by Feynman [22] has so far not been observed. We present the experimental realization of such a single quantum particle walking in a one-dimensional (1D) lattice in position space. This basic example of a walk provides all of the relevant features necessary to understand the fundamental properties and differences of the quantum and classical regimes. For example, the atomic wave func- * Electronic address: karski@uni-bonn.de † Electronic address: widera@uni-bonn.de tion resulting from a quantum walk exhibit...

We study in detail the mechanisms causing dephasing of hyperfine coherences of cesium atoms confined by a far off-resonant standing wave optical dipole trap [S. Kuhr et al., Phys. Rev. Lett. 91, 213002 (2003)]. Using Ramsey spectroscopy and spin echo techniques, we measure the reversible and irreversible dephasing times of the ground state coherences. We present an analytical model to interpret the experimental data and identify the homogeneous and inhomogeneous dephasing mechanisms. Our scheme to prepare and detect the atomic hyperfine state is applied at the level of a single atom as well as for ensembles of up to 50 atoms.

We report on a stringent test of the non-classicality of the motion of a massive quantum particle, which propagates on a discrete lattice. Measuring temporal correlations of the position of single atoms performing a quantum walk, we observe a 6 σ violation of the Leggett-Garg inequality. Our results rigorously excludes (i.e. falsifies) any explanation of quantum transport based on classical, well-defined trajectories. We use so-called ideal negative measurements an essential requisite for any genuine Leggett-Garg test to acquire information about the atom's position, yet avoiding any direct interaction with it. The interaction-free measurement is based on a novel atom transport system, which allows us to directly probe the absence rather than the presence of atoms at a chosen lattice site. Beyond the fundamental aspect of this test, we demonstrate the application of the Leggett-Garg correlation function as a witness of quantum superposition. We here employ the witness to discriminate different types of walks spanning from merely classical to wholly quantum dynamics. Subject Areas: Quantum Physics, Atomic and Molecular Physics Keywords: Leggett-Garg inequality, ideal negative measurements, quantum walks, quantum witnesses arXiv:1404.3912v2 [quant-ph] 10 Feb 2015 ≈ 0.2 %. Hence, we quantify the relative frequency of non-ideal negative measurements with ζ = 1 %. Along the lines of Ref. 26, the correlation function K measured in our experiment can be decomposed as K = 1 + (1 − ζ)K ideal 23 + ζK corrupt 23 − K 13 , where K ideal 23 and K corrupt 23denote the correlation function Q(t 3 )Q(t 2 ) which has been measured with an ideal negative measurement Q(t 2 ) and with a corrupted one, respectively.

We report on the experimental realization of electric quantum walks, which mimic the effect of an electric field on a charged particle in a lattice. Starting from a textbook implementation of discrete-time quantum walks, we introduce an extra operation in each step to implement the effect of the field. The recorded dynamics of such a quantum particle exhibits features closely related to Bloch oscillations and interband tunneling. In particular, we explore the regime of strong fields, demonstrating contrasting quantum behaviors: quantum resonances versus dynamical localization depending on whether the accumulated Bloch phase is a rational or irrational fraction of 2π.

We present the design of a diffraction limited, long working distance monochromatic objective lens for efficient light collection. Consisting of four spherical lenses, it has a numerical aperture of 0.29, an effective focal length of 36 mm and a working distance of 36.5 mm. This inexpensive system allows us to detect 8 · 10 4 fluorescence photons per second from a single cesium atom stored in a magneto-optical trap.

The strong evanescent field around ultrathin unclad optical fibers bears a high potential for detecting, trapping, and manipulating cold atoms. Introducing such a fiber into a cold-atom cloud, we investigate the interaction of a small number of cold cesium atoms with the guided fiber mode and with the fiber surface. Using high resolution spectroscopy, we observe and analyze light-induced dipole forces, van der Waals interaction, and a significant enhancement of the spontaneous emission rate of the atoms. The latter can be assigned to the modification of the vacuum modes by the fiber.

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