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Geometrically frustrated many-particle quantum systems are notoriously hard to study numerically but are of profound interest because of their unusual properties and emergent phenomena. In these systems energetic constraints cannot be minimized simultaneously, leading to large ground-state degeneracy and a variety of exotic quantum phases. Here, we present a platform that enables unprecedentedly detailed experimental exploration of geometrically frustrated electronic systems on lattices with triangular geometry. We demonstrate the first realization of triangular atomic Hubbard systems, directly image Mott insulators in the triangular geometry with single-atom and single-site resolution, and measure antiferromagnetic spin-spin correlations for all nearest neighbors allowing for thermometry. This platform provides a powerful new approach for studying exotic quantum magnetism and direct detection of quantum spin liquid signatures in Hubbard systems., Comment: 13 pages, 10 figures

To support the understanding of declarative probabilistic programming languages, we introduce a lambda-calculus with a fair binary probabilistic choice that chooses between its arguments with equal probability. The reduction strategy of the calculus is a call-by-need strategy that performs lazy evaluation and implements sharing by recursive let-expressions. Expected convergence of expressions is the limit of the sum of all successful reduction outputs weighted by their probability. We use contextual equivalence as program semantics: two expressions are contextually equivalent if and only if the expected convergence of the expressions plugged into any program context is always the same. We develop and illustrate techniques to prove equivalences including a context lemma, two derived criteria to show equivalences and a syntactic diagram-based method. This finally enables us to show correctness of a large set of program transformations with respect to the contextual equivalence., Comment: 18 pages, 11 figures

The suppression of antiferromagnetic ordering in geometrically frustrated Hubbard models leads to a variety of exotic quantum phases including quantum spin liquids and chiral states. Here, we focus on the Hubbard model on one of the simplest frustrated lattice geometries, a triangular lattice. Motivated by the recent realization of ultracold fermionic atoms in triangular optical lattices, we study the properties of the triangular-lattice Hubbard model through a Numerical Linked-Cluster Expansion algorithm. We investigate the Mott insulator transition finding a critical interaction $U_c/t = 7.0(2)$ and use spatial two- and three-point correlation functions to explore doped and imbalanced systems. Our results demonstrate that many interesting features occur at temperatures previously obtained for ultracold fermions in optical lattices and are accessible by upcoming experiments. Our calculations will be helpful for thermometry in ultracold atom quantum simulators and can guide experimental searches for exotic quantum phases in atomic triangular Hubbard quantum simulators., Comment: 9 pages, 10 figures

The distinct structures of contact ion pairs in water are identified. Nonlinear infrared (IR) spectroscopy and theoretical calculations allow for the separation and assignment of spectral features and interactions.

We discern hydration geometries around the sugar-phosphate-backbone of an RNA double helix at the molecular level. RNA disordering upon melting is connected with a transition from ordered water structures towards local phosphate group hydration shells.

Electric interactions have a strong impact on the structure and dynamics of biomolecules in their native water environment. Given the variety of water arrangements in hydration shells and the femto- to subnanosecond time range of structural fluctuations, there is a strong quest for sensitive noninvasive probes of local electric fields. The stretching vibrations of phosphate groups, in particular the asymmetric (PO2)- stretching vibration {\nu}AS(PO2)-, allow for a quantitative mapping of dynamic electric fields in aqueous environments via a field-induced redshift of their transition frequencies and concomitant changes of vibrational line shapes. We present a systematic study of {\nu}AS(PO2)- excitations in molecular systems of increasing complexity, including dimethyl phosphate (DMP), short DNA and RNA duplex structures, and transfer RNA (tRNA) in water. A combination of linear infrared absorption, two-dimensional infrared (2D-IR) spectroscopy, and molecular dynamics (MD) simulations gives quantitative insight in electric-field tuning rates of vibrational frequencies, electric field and fluctuation amplitudes, and molecular interaction geometries. Beyond neat water environments, the formation of contact ion pairs of phosphate groups with Mg2+ ions is demonstrated via frequency upshifts of the {\nu}AS(PO2)- vibration, resulting in a distinct vibrational band. The frequency positions of contact geometries are determined by an interplay of attractive electric and repulsive exchange interactions., Comment: 9 pages, 6 figures

Ions interacting with hydrated RNA play a central role in defining its secondary and tertiary structure. While spatial arrangements of ions, water molecules, and phosphate groups have been inferred from X-ray studies, the role of electrostatic and other noncovalent interactions in stabilizing compact folded RNA structures is not fully understood at the molecular level. Here, we demonstrate that contact ion pairs of magnesium (Mg2+) and phosphate groups embedded in local water shells stabilize the tertiary equilibrium structure of transfer RNA (tRNA). Employing dialyzed tRNAPhe from yeast and tRNA from Escherichia coli, we follow the population of Mg2+ sites close to phosphate groups of the ribose-phosphodiester backbone step by step, combining linear and nonlinear infrared spectroscopy of phosphate vibrations with molecular dynamics simulations and ab initio vibrational frequency calculations. The formation of up to six Mg2+/phosphate contact pairs per tRNA and local field-induced reorientations of water molecules balance the phosphate-phosphate repulsion in nonhelical parts of tRNA, thus stabilizing the folded structure electrostatically. Such geometries display limited sub-picosecond fluctuations in the arrangement of water molecules and ion residence times longer than 1 {\mu}s. At higher Mg2+ excess, the number of contact ion pairs per tRNA saturates around 6 and weakly interacting ions prevail. Our results suggest a predominance of contact ion pairs over long-range coupling of the ion atmosphere and the biomolecule in defining and stabilizing the tertiary structure of tRNA., Comment: 7 pages, 4 figures

In order to understand the relative expressive power of larger concurrent programming languages, we analyze translations of small process calculi which model the communication and synchronization of concurrent processes. The source language SYNCSIMPLE is a minimalistic model for message passing concurrency while the target language LOCKSIMPLE is a minimalistic model for shared memory concurrency. The former is a calculus with synchronous communication of processes, while the latter has synchronizing mutable locations - called locks - that behave similarly to binary semaphores. The criteria for correctness of translations is that they preserve and reflect may-termination and must-termination of the processes. We show that there is no correct compositional translation from SYNCSIMPLE to LOCKSIMPLE that uses one or two locks, independent from the initialisation of the locks. We also show that there is a correct translation that uses three locks. Also variants of the locks are taken into account with different blocking behavior., Comment: In Proceedings EXPRESS/SOS 2021, arXiv:2108.09624. Author version at 2107.14651

In order to understand the relative expressive power of larger concurrent programming languages, we analyze translations of small process calculi which model the communication and synchronization of concurrent processes. The source language SYNCSIMPLE is a minimalistic model for message passing concurrency while the target language LOCKSIMPLE is a minimalistic model for shared memory concurrency. The former is a calculus with synchronous communication of processes, while the latter has synchronizing mutable locations -- called locks -- that behave similarly to binary semaphores. The criteria for correctness of translations is that they preserve and reflect may-termination and must-termination of the processes. We show that there is no correct compositional translation from SYNCSIMPLE to LOCKSIMPLE that uses one or two locks, independent from the initialisation of the locks. We also show that there is a correct translation that uses three locks. Also variants of the locks are taken into account with different blocking behavior.

Quantum gas microscopes have expanded the capabilities of quantum simulation of Hubbard models by enabling the study of spatial spin and density correlations in square lattices. However, quantum gas microscopes have not been realized for fermionic atoms in frustrated geometries. Here, we demonstrate the single-atom resolved imaging of ultracold fermionic $^{6}$Li atoms in a triangular optical lattice with a lattice constant of 1003 nm. The optical lattice is formed by a recycled narrow-linewidth, high-power laser combined with a light sheet to allow for Raman sideband cooling on the $D_1$ line. We optically resolve single atoms on individual lattice sites using a high-resolution objective to collect scattered photons while cooling them close to the two-dimensional ground vibrational level in each lattice site. By reconstructing the lattice occupation, we measure an imaging fidelity of ~98%. Our new triangular lattice microscope platform for fermions clears the path for studying spin-spin correlations, entanglement and dynamics of geometrically frustrated Hubbard systems which are expected to exhibit exotic emergent phenomena including spin liquids and kinetic frustration., Comment: 9 pages, 5 figures, comments welcome!

A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in nondeterministic polynomial time. We also explore specializations like nominal letrec-matching for expressions, for DAGs, and for garbage-free expressions and determine their complexity. We also provide a nominal unification algorithm for higher-order expressions with recursive let and atom-variables, where we show that it also runs in nondeterministic polynomial time. In addition we prove that there is a guessing strategy for nominal unification with letrec and atom-variable that is a trade-off between exponential growth and non-determinism. Nominal matching with variables representing partial letrec-environments is also shown to be in NP., Comment: 37 pages, 9 figures, This paper is an extended version of the conference publication: Manfred Schmidt-Schau{\ss} and Temur Kutsia and Jordi Levy and Mateu Villaret and Yunus Kutz, Nominal Unification of Higher Order Expressions with Recursive Let, LOPSTR-16, Lecture Notes in Computer Science 10184, Springer, p 328 -344, 2016. arXiv admin note: text overlap with arXiv:1608.03771

We induce strong non-local interactions in a 2D Fermi gas in an optical lattice using Rydberg dressing. The system is approximately described by a $t-V$ model on a square lattice where the fermions experience isotropic nearest-neighbor interactions and are free to hop only along one direction. We measure the interactions using many-body Ramsey interferometry and study the lifetime of the gas in the presence of tunneling, finding that tunneling does not reduce the lifetime. To probe the interplay of non-local interactions with tunneling, we investigate the short-time relaxation dynamics of charge density waves in the gas. We find that strong nearest-neighbor interactions slow down the relaxation. Our work opens the door for quantum simulations of systems with strong non-local interactions such as extended Fermi-Hubbard models., Comment: 12 pages, 6 figures in main text, 3 figures in appendix

Comparison of concurrent programming languages and correctness of program transformations in concurrency are the focus of this research. As criterion we use contextual semantics adapted to concurrency, where may -- as well as should -- convergence are observed. We investigate the relation between the synchronous pi-calculus and a core language of Concurrent Haskell (CH). The contextual semantics is on the one hand forgiving with respect to the details of the operational semantics, and on the other hand implies strong requirements for the interplay between the processes after translation. Our result is that CH embraces the synchronous pi-calculus. Our main task is to find and prove correctness of encodings of pi-calculus channels by CH's concurrency primitives, which are MVars. They behave like (blocking) 1-place buffers modelling the shared-memory. The first developed translation uses an extra private MVar for every communication.We also automatically generate and check potentially correct translations that reuse the MVars where one MVar contains the message and two additional MVars for synchronization are used to model the synchronized communication of a single channel in the pi-calculus.Our automated experimental results lead to the conjecture that one additional MVar is insufficient., Comment: In Proceedings EXPRESS/SOS 2020, arXiv:2008.12414

The wake following a vessel in water is a signature interference effect of moving bodies, and, as described by Lord Kelvin, is contained within a constant universal angle. However, wakes may accompany different kinds of moving disturbances in other situations and even in lattice systems. Here, we investigate the effect of moving disturbances on a Fermi lattice gas of ultracold atoms and analyze the novel types of wake patterns that may occur. We show how at half-filling, the wake angles are dominated by the ratio of the hopping energy to the velocity of the disturbance and on the angle of motion relative to the lattice direction. Moreover, we study the difference between wakes left behind a moving particle detector versus that of a moving potential or a moving particle extractor. We show that these scenarios exhibit dramatically different behavior at half-filling, with the "measurement wake" following an idealized detector vanishing, though the motion of the detector does still leaves a trace through a "fluctuation wake." Finally, we discuss the experimental requirements to observe our predictions in ultracold fermionic atoms in optical lattices.

The extent of contact and solvent shared ion pairs of phosphate groups with Na$^+$, Ca$^{2+}$ and Mg$^{2+}$ ions in aqueous environment and their relevance for the stability of polyanionic DNA and RNA structures is highly debated. Employing the asymmetric phosphate stretching vibration of dimethyl phosphate (DMP), a model system of the sugar-phosphate backbone of DNA and RNA, we present linear infrared, femtosecond infrared pump-probe and absorptive 2D-IR spectra that report on contact ion pair formation via the presence of blue shifted spectral signatures. Compared to the linear infrared spectra, the nonlinear spectra reveal contact ion pairs with increased sensitivity because the spectra accentuate differences in peak frequency, transition dipole moment strength, and excited state lifetime. The experimental results are corroborated by long time scale MD simulations, benchmarked by density functional simulations on phosphate-ion-water clusters. The microscopic interpretation reveals subtle structural differences of ion pairs formed by the phosphate group and the ions Na$^+$, Ca$^{2+}$ and Mg$^{2+}$. Intricate properties of the solvation shell around the phosphate group and the ion are essential to explain the experimental observations. The present work addresses a challenging to probe topic with the help of a model system and establishes new experimental data of contact ion pair formation, thereby underlining the potential of nonlinear 2D-IR spectroscopy as an analytical probe of phosphate-ion interactions in complex biological systems.

The water hydration shell has decisive impact on the structural and functional properties of RNA. Changes of RNA structure upon melting and in biochemical processes are accompanied by a change of hydration patterns, a process which is barely characterized. To discern hydration geometries around the backbone phosphate groups of an RNA double helix at the molecular level, we combine two-dimensional infrared spectroscopy of phosphate vibrations with theoretical simulations. There are three distinct coexisting hydration motifs of the RNA A-helix, an ordered chain-like arrangement of water molecules with links between neighboring phosphate groups, separate local hydration shells of up to six water molecules, and hydrated phosphate/counterion contact pairs. RNA disordering upon melting is connected with a transition from predominant ordered water structures to local hydration shells around phosphate units. Structural fluctuations are dominated by librational water motions occurring on a 300 fs time scale, without exchange between hydration motifs.

The interaction of phosphate groups with ions in an aqueous environment has a strong impact on the structure and folding processes of DNA and RNA. The dynamic variety of ionic arrangements, including both contact pairs and water separated ions, and the molecular coupling mechanisms are far from being understood. In a combined experimental and theoretical approach, we address the properties of contact ion pairs of the prototypical system dimethyl-phosphate with Na$^+$, Ca$^{2+}$, and Mg$^{2+}$ ions in water. Linear and femtosecond two-dimensional infrared (2D-IR) spectroscopy of the asymmetric (PO$_2$)$^-$ stretching vibration separates and characterizes the different species via their blue-shifted vibrational signatures and 2D-IR lineshapes. Phosphate-magnesium contact pairs stand out as the most compact geometry while the contact pairs with Ca$^{2+}$ and Na$^+$ display a wider structural variation. Microscopic density functional theory simulations rationalize the observed frequency shifts and reveal distinct differences between the contact geometries.

This paper is a contribution to exploring and analyzing space-improvements in concurrent programming languages, in particular in the functional process-calculus CHF. Space-improvements are defined as a generalization of the corresponding notion in deterministic pure functional languages. The main part of the paper is the O(n*log n) algorithm SpOptN for offline space optimization of several parallel independent processes. Applications of this algorithm are: (i) affirmation of space improving transformations for particular classes of program transformations; (ii) support of an interpreter-based method for refuting space-improvements; and (iii) as a stand-alone offline-optimizer for space (or similar resources) of parallel processes., Comment: In Proceedings WPTE 2018, arXiv:1902.07818

Charge transport is a revealing probe of the quantum properties of materials. Strong interactions can blur charge carriers resulting in a poorly understood "quantum soup". Here we study the conductivity of the Fermi-Hubbard model, a testing ground for strong interaction physics, in a clean quantum system - ultracold $^6$Li in a 2D optical lattice. We determine the charge diffusion constant in our system by measuring the relaxation of an imposed density modulation and modeling its decay hydrodynamically. The diffusion constant is converted to a resistivity, which exhibits a linear temperature dependence and exceeds the Mott-Ioffe-Regel limit, two characteristic signatures of a bad metal. The techniques we develop here may be applied to measurements of other transport quantities, including the optical conductivity and thermopower.

We explore space improvements in LRP, a polymorphically typed call-by-need functional core language. A relaxed space measure is chosen for the maximal size usage during an evaluation. It abstracts from the details of the implementation via abstract machines, but it takes garbage collection into account and thus can be seen as a realistic approximation of space usage. The results are: a context lemma for space improving translations and for space equivalences, all but one reduction rule of the calculus are shown to be space improvements, and for the exceptional one we show bounds on the space increase. Several further program transformations are shown to be space improvements or space equivalences in particular the translation into machine expressions is a space equivalence. We also classify certain space-worsening transformations as space-leaks or as space-safe. These results are a step forward in making predictions about the change in runtime space behavior of optimizing transformations in call-by-need functional languages., Comment: In Proceedings WPTE 2017, arXiv:1802.05862

Simulating the real-time evolution of quantum spin systems far out of equilibrium poses a major theoretical challenge, especially in more than one dimension. We experimentally explore the dynamics of a two-dimensional Ising spin system with transverse and longitudinal fields as we quench it across a quantum phase transition from a paramagnet to an antiferromagnet. We realize the system with a near unit-occupancy atomic array of over 200 atoms obtained by loading a spin-polarized band insulator of fermionic lithium into an optical lattice and induce short-range interactions by direct excitation to a low-lying Rydberg state. Using site-resolved microscopy, we probe the correlations in the system after a sudden quench from the paramagnetic state and compare our measurements to exact calculations in the regime where it is possible. We achieve many-body states with longer-range antiferromagnetic correlations by implementing a near-adiabatic quench and study the buildup of correlations as we cross the quantum phase transition at different rates.

Finite-range interacting spin models are the simplest models to study the effect of beyond nearest-neighbour interactions and access new effects caused by the range of the interactions. Recent experiments have reached the regime of dominant interactions in Ising quantum magnets via optical coupling of trapped neutral atoms to Rydberg states. This approach allows for the tunability of all relevant terms in an Ising Hamiltonian with $1/r^6$ interactions in a transverse and longitudinal field. This review summarizes the recent progress of these implementations in Rydberg lattices with site-resolved detection. The strong correlations in this quantum Ising model have been observed in several experiments up to the point of crystallization. In systems with a diameter small compared to the Rydberg blockade radius, the number of excitations is maximally one in the so-called superatom regime., Comment: 18 pages, 15 figures

The attractive Fermi-Hubbard model is the simplest theoretical model for studying pairing and superconductivity of fermions on a lattice. Although its s-wave pairing symmetry excludes it as a microscopic model for high-temperature superconductivity, it exhibits much of the relevant phenomenology, including a short-coherence length at intermediate coupling and a pseudogap regime with anomalous properties. Here we study an experimental realization of this model using a two-dimensional (2D) atomic Fermi gas in an optical lattice. Our site-resolved measurements on the normal state reveal checkerboard charge-density-wave correlations close to half-filling. A "hidden" SU(2) pseudo-spin symmetry of the Hubbard model at half-filling guarantees superfluid correlations in our system, the first evidence for such correlations in a single-band Hubbard system of ultracold fermions. Compared to the paired atom fraction, we find the charge-density-wave correlations to be a much more sensitive thermometer, useful for optimizing cooling into superfluid phases in future experiments.

We present an implementation of an interpreter LRPi for the call-by-need calculus LRP, based on a variant of Sestoft's abstract machine Mark 1, extended with an eager garbage collector. It is used as a tool for exact space usage analyses as a support for our investigations into space improvements of call-by-need calculi., Comment: In Proceedings WPTE 2016, arXiv:1701.00233

Understanding the magnetic response of the normal state of the cuprates is considered a key piece in solving the puzzle of their high-temperature superconductivity. The essential physics of these materials is believed to be captured by the Fermi-Hubbard model, a minimal model that has been realized with cold atoms in optical lattices. Here we report on site-resolved measurements of the Fermi-Hubbard model in a spin-imbalanced atomic gas, allowing us to explore the response of the system to large effective magnetic fields. We observe short-range canted antiferromagnetism at half-filling with stronger spin correlations in the direction orthogonal to the magnetization, in contrast with the spin-balanced case where identical correlations are measured for any projection of the pseudospin. The rotational anisotropy of the spin correlators is found to increase with polarization and with distance between the spins. Away from half-filling, the polarization of the gas exhibits non-monotonic behavior with doping for strong interactions, resembling the behavior of the magnetic susceptibility in the cuprates. We compare our measurements to predictions from Determinantal Quantum Monte Carlo (DQMC) and Numerical Linked Cluster Expansion (NLCE) algorithms and find good agreement. Calculations on the doped system are near the limits of these techniques, illustrating the value of cold atom quantum simulations for studying strongly-correlated materials.

We present the development of the first ultrafast transmission electron microscope (UTEM) driven by localized photoemission from a field emitter cathode. We describe the implementation of the instrument, the photoemitter concept and the quantitative electron beam parameters achieved. Establishing a new source for ultrafast TEM, the G\"ottingen UTEM employs nano-localized linear photoemission from a Schottky emitter, which enables operation with freely tunable temporal structure, from continuous wave to femtosecond pulsed mode. Using this emission mechanism, we achieve record pulse properties in ultrafast electron microscopy of 9 {\AA} focused beam diameter, 200 fs pulse duration and 0.6 eV energy width. We illustrate the possibility to conduct ultrafast imaging, diffraction, holography and spectroscopy with this instrument and also discuss opportunities to harness quantum coherent interactions between intense laser fields and free electron beams.

A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in non-deterministic polynomial time. We also explore specializations like nominal letrec-matching for plain expressions and for DAGs and determine the complexity of corresponding unification problems., Comment: Pre-proceedings paper presented at the 26th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2016), Edinburgh, Scotland UK, 6-8 September 2016 (arXiv:1608.02534)

One fundamental assumption in statistical physics is that generic closed quantum many-body systems thermalize under their own dynamics. Recently, the emergence of many-body localized systems has questioned this concept, challenging our understanding of the connection between statistical physics and quantum mechanics. Here we report on the observation of a many-body localization transition between thermal and localized phases for bosons in a two-dimensional disordered optical lattice. With our single site resolved measurements we track the relaxation dynamics of an initially prepared out-of-equilibrium density pattern and find strong evidence for a diverging length scale when approaching the localization transition. Our experiments mark the first demonstration and in-depth characterization of many-body localization in a regime not accessible with state-of-the-art simulations on classical computers., Comment: 10 pages and 9 figures

We study a two-component quasi-two-dimensional Fermi gas with imbalanced spin populations. We probe the gas at different interaction strengths and polarizations by measuring the density of each spin component in the trap and the pair momentum distribution after time of flight. For a wide range of experimental parameters, we observe in-trap phase separation characterized by the appearance of a spin-balanced condensate surrounded by a polarized gas. Our momentum space measurements indicate pair condensation in the imbalanced gas even for large polarizations where phase separation vanishes, pointing to the presence of a polarized pair condensate. Our observation of zero momentum pair condensates in 2D spin-imbalanced gases opens the way to explorations of more exotic superfluid phases that occupy a large part of the phase diagram in lower dimensions.

Ultracold atoms are an ideal platform to study strongly correlated phases of matter in and out of equilibrium. Much of the experimental progress in this field crucially relies on the control of the contact interaction between two atoms. Control of strong long-range interactions between distant ground state atoms has remained a long standing goal, opening the path towards the study of fundamentally new quantum many-body systems including frustrated or topological magnets and supersolids. Optical dressing of ground state atoms by near-resonant laser coupling to Rydberg states has been proposed as a versatile method to engineer such interactions. However, up to now the great potential of this approach for interaction control in a many-body setting has eluded experimental confirmation. Here we report the realisation of coherent Rydberg-dressing in an ultracold atomic lattice gas and directly probe the induced interaction potential using an interferometric technique with single atom sensitivity. We use this approach to implement a two-dimensional synthetic spin lattice and demonstrate its versatility by tuning the range and anisotropy of the effective spin interactions. Our measurements are in remarkable agreement with exact solutions of the many-body dynamics, providing further evidence for the high degree of accurate interaction control in these systems. Finally, we identify a collective many-body decay process, and discuss possible routes to overcome this current limitation of coherence times. Our work marks the first step towards the use of laser-controlled Rydberg interactions for the study of exotic quantum magnets in optical lattices., Comment: 18 pages, 13 figures; Version initially submitted to Nature Physics

Entanglement is an essential property of quantum many-body systems. However, its local detection is challenging and was so far limited to spin degrees of freedom in ion chains. Here we measure entanglement between the spins of atoms located on two lattice sites in a one-dimensional Bose-Hubbard chain which features both local spin- and particle-number fluctuations. Starting with an initially localized spin impurity, we observe an outwards propagating entanglement wave and show quantitatively how entanglement in the spin sector rapidly decreases with increasing particle-number fluctuations in the chain., Comment: 6 pages, 4 figures

Strong interactions can amplify quantum effects such that they become important on macroscopic scales. Controlling these coherently on a single particle level is essential for the tailored preparation of strongly correlated quantum systems and opens up new prospects for quantum technologies. Rydberg atoms offer such strong interactions which lead to extreme nonlinearities in laser coupled atomic ensembles. As a result, multiple excitation of a Micrometer sized cloud can be blocked while the light-matter coupling becomes collectively enhanced. The resulting two-level system, often called "superatom", is a valuable resource for quantum information, providing a collective Qubit. Here we report on the preparation of two orders of magnitude scalable superatoms utilizing the large interaction strength provided by Rydberg atoms combined with precise control of an ensemble of ultracold atoms in an optical lattice. The latter is achieved with sub shot noise precision by local manipulation of a two-dimensional Mott insulator. We microscopically confirm the superatom picture by in-situ detection of the Rydberg excitations and observe the characteristic square root scaling of the optical coupling with the number of atoms. Furthermore, we verify the presence of entanglement in the prepared states and demonstrate the coherent manipulation of the superatom. Finally, we investigate the breakdown of the superatom picture when two Rydberg excitations are present in the system, which leads to dephasing and a loss of coherence., Comment: 7 pages, 5 figures

This paper shows equivalence of several versions of applicative similarity and contextual approximation, and hence also of applicative bisimilarity and contextual equivalence, in LR, the deterministic call-by-need lambda calculus with letrec extended by data constructors, case-expressions and Haskell's seq-operator. LR models an untyped version of the core language of Haskell. The use of bisimilarities simplifies equivalence proofs in calculi and opens a way for more convenient correctness proofs for program transformations. The proof is by a fully abstract and surjective transfer into a call-by-name calculus, which is an extension of Abramsky's lazy lambda calculus. In the latter calculus equivalence of our similarities and contextual approximation can be shown by Howe's method. Similarity is transferred back to LR on the basis of an inductively defined similarity. The translation from the call-by-need letrec calculus into the extended call-by-name lambda calculus is the composition of two translations. The first translation replaces the call-by-need strategy by a call-by-name strategy and its correctness is shown by exploiting infinite trees which emerge by unfolding the letrec expressions. The second translation encodes letrec-expressions by using multi-fixpoint combinators and its correctness is shown syntactically by comparing reductions of both calculi. A further result of this paper is an isomorphism between the mentioned calculi, which is also an identity on letrec-free expressions., Comment: 50 pages, 11 figures