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Quantum field theory is a powerful tool to describe the relevant physics governing complex quantum many-body systems. Here we develop a general pathway to extract the irreducible building blocks of quantum field theoretical descriptions and its parameters purely from experimental data. This is accomplished by extracting the one-particle irreducible (1PI) vertices from which one can construct all observables. To match the capabilities of experimental techniques used in quantum simulation experiments, our approach employs a formulation of quantum field theory based on equal-time correlation functions only. We illustrate our procedure by applying it to the quantum sine-Gordon model in thermal equilibrium. The theoretical foundations are illustrated by estimating the irreducible vertices at equal times both analytically and using numerical simulations. We then demonstrate explicitly how to extract these quantities from an experiment where we quantum simulate the sine-Gordon model by two tunnel-coupled superfluids. We extract the full two-point function and the interaction vertex (four-point function) and their variation with momentum, encoding the `running' of the couplings. The measured 1PI vertices are compared to the theoretical estimates, verifying our procedure. Our work opens new ways of addressing fundamental questions in quantum field theory, which are relevant in high-energy and condensed matter physics, and in taking quantum phenomena from fundamental science to practical technology.

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Phase transitions are driven by collective fluctuations of a system’s constituents that emerge at a critical point¹. This mechanism has been extensively explored for classical and quantum systems in equilibrium, whose critical behaviour is described by the general theory of phase transitions. Recently, however, fundamentally distinct phase transitions have been discovered for out-of-equilibrium quantum systems, which can exhibit critical behaviour that defies this description and is not well understood¹. A paradigmatic example is the many-body localization (MBL) transition, which marks the breakdown of thermalization in an isolated quantum many-body system as its disorder increases beyond a critical value2–11. Characterizing quantum critical behaviour in an MBL system requires probing its entanglement over space and time4,5,7, which has proved experimentally challenging owing to stringent requirements on quantum state preparation and system isolation. Here we observe quantum critical behaviour at the MBL transition in a disordered Bose–Hubbard system and characterize its entanglement via its multi-point quantum correlations. We observe the emergence of strong correlations, accompanied by the onset of anomalous diffusive transport throughout the system, and verify their critical nature by measuring their dependence on the system size. The correlations extend to high orders in the quantum critical regime and appear to form via a sparse network of many-body resonances that spans the entire system12,13. Our results connect the macroscopic phenomenology of the transition to the system’s microscopic structure of quantum correlations, and they provide an essential step towards understanding criticality and universality in non-equilibrium systems1,7,13.

The exploration of quantum phenomena in a curved spacetime is an emerging interdisciplinary area at the interface between general relativity1–4, thermodynamics4–6 and quantum information7,8. One famous prediction in this field is Unruh thermal radiation³—the manifestation of thermal radiation from a Minkowski vacuum when viewed in an accelerating reference frame. Here, we report the experimental observation of a matter field with thermal fluctuations that agree with Unruh’s predictions. The matter field is generated within a framework for the simulation of quantum physics in a non-inertial frame, based on Bose–Einstein condensates that are parametrically modulated⁹ to make their evolution replicate the frame transformation. We further observe long-range phase coherence and temporal reversal of the matter-wave radiation, hallmarks that distinguish Unruh radiation from its classical counterpart. Our demonstration offers a new avenue for the investigation of the dynamics of quantum many-body systems in a curved spacetime.

Quantum phase transitions (QPTs) involve transformations between different states of matter that are driven by quantum fluctuations1. These fluctuations play a dominant part in the quantum critical region surrounding the transition point, where the dynamics is governed by the universal properties associated with the QPT. Although time-dependent phenomena associated with classical, thermally driven phase transitions have been extensively studied in systems ranging from the early Universe to Bose–Einstein condensates2–5, understanding critical real-time dynamics in isolated, non-equilibrium quantum systems remains a challenge6. Here we use a Rydberg atom quantum simulator with programmable interactions to study the quantum critical dynamics associated with several distinct QPTs. By studying the growth of spatial correlations when crossing the QPT, we experimentally verify the quantum Kibble–Zurek mechanism (QKZM)7–9 for an Ising-type QPT, explore scaling universality and observe corrections beyond QKZM predictions. This approach is subsequently used to measure the critical exponents associated with chiral clock models10,11, providing new insights into exotic systems that were not previously understood and opening the door to precision studies of critical phenomena, simulations of lattice gauge theories12,13 and applications to quantum optimization14,15. A Rydberg atom quantum simulator with programmable interactions is used to experimentally verify the quantum Kibble–Zurek mechanism through the growth of spatial correlations during quantum phase transitions.

Understanding the behaviour of isolated quantum systems far from equilibrium and their equilibration is one of the most pressing problems in quantum many-body physics1,2. There is strong theoretical evidence that sufficiently far from equilibrium a wide variety of systems—including the early Universe after inflation3–6, quark–gluon matter generated in heavy-ion collisions7–9, and cold quantum gases4,10–14—exhibit universal scaling in time and space during their evolution, independent of their initial state or microscale properties. However, direct experimental evidence is lacking. Here we demonstrate universal scaling in the time-evolving momentum distribution of an isolated, far-from-equilibrium, one-dimensional Bose gas, which emerges from a three-dimensional ultracold Bose gas by means of a strong cooling quench. Within the scaling regime, the time evolution of the system at low momenta is described by a time-independent, universal function and a single scaling exponent. The non-equilibrium scaling describes the transport of an emergent conserved quantity towards low momenta, which eventually leads to the build-up of a quasi-condensate. Our results establish universal scaling dynamics in an isolated quantum many-body system, which is a crucial step towards characterizing time evolution far from equilibrium in terms of universality classes. Universality would open the possibility of using, for example, cold-atom set-ups at the lowest energies to simulate important aspects of the dynamics of currently inaccessible systems at the highest energies, such as those encountered in the inflationary early Universe.

Predicting the dynamics of quantum systems far from equilibrium represents one of the most challenging problems in theoretical many-body physics1,2. While the evolution of a many-body system is in general intractable in all its details, relevant observables can become insensitive to microscopic system parameters and initial conditions. This is the basis of the phenomenon of universality. Far from equilibrium, universality is identified through the scaling of the spatio-temporal evolution of the system, captured by universal exponents and functions. Theoretically, this has been studied in examples as different as the reheating process in inflationary Universe cosmology3,4, the dynamics of nuclear collision experiments described by quantum chromodynamics5,6, and the post-quench dynamics in dilute quantum gases in non-relativistic quantum field theory7–11. However, an experimental demonstration of such scaling evolution in space and time in a quantum many-body system has been lacking. Here we observe the emergence of universal dynamics by evaluating spatially resolved spin correlations in a quasi-one-dimensional spinor Bose–Einstein condensate12–16. For long evolution times we extract the scaling properties from the spatial correlations of the spin excitations. From this we find the dynamics to be governed by an emergent conserved quantity and the transport of spin excitations towards low momentum scales. Our results establish an important class of non-stationary systems whose dynamics is encoded in time-independent scaling exponents and functions, signalling the existence of non-thermal fixed points10,17,18. We confirm that the non-thermal scaling phenomenon involves no fine-tuning of parameters, by preparing different initial conditions and observing the same scaling behaviour. Our analogue quantum simulation approach provides the basis with which to reveal the underlying mechanisms and characteristics of non-thermal universality classes. One may use this universality to learn, from experiments with ultracold gases, about fundamental aspects of dynamics studied in cosmology and quantum chromodynamics.

Seeing patterns in atomic jets
Atomic interactions in a Bose-Einstein condensate (BEC) can lead to complex collective behavior. Experimentally, these interactions are often tuned by varying an external magnetic field. Feng et al. modulated the interaction among cesium atoms in a BEC. The collisions between atoms exposed to the modulated field sent the atoms flying out of the condensate in jets of seemingly random directions. A pattern-recognition technique revealed that certain directions were associated with particularly large numbers of scattered atoms. The pattern of the scattering maxima could be attributed to secondary collisions.
Science , this issue p. 521

We study the dynamics of a supersonically expanding ring-shaped Bose-Einstein condensate both experimentally and theoretically. The expansion redshifts long-wavelength excitations, as in an expanding universe. After expansion, energy in the radial mode leads to the production of bulk topological excitations -- solitons and vortices -- driving the production of a large number of azimuthal phonons and, at late times, causing stochastic persistent currents. This reheating of the condensate is reminiscent of the presumed reheating of the universe after inflation.

Recurring coherence
A finite isolated system should return almost to its initial state if it evolves for long enough. For a large system, “long enough” is often unfeasibly long. Rauer et al. found just the right conditions to observe the recurrence of the initial state in a system of two one-dimensional superfluids with thousands of atoms in each. The superfluids were initially coupled—locking their quantum mechanical phases together—and then allowed to evolve independently. After the uncoupling, the researchers observed their phases regaining coherence two more times.
Science , this issue p. 307

The description of the non-equilibrium dynamics of isolated quantum many-body systems within the framework of statistical mechanics is a fundamental open question. Conventional thermodynamical ensembles fail to describe the large class of systems that exhibit nontrivial conserved quantities, and generalized ensembles have been predicted to maximize entropy in these systems. We show experimentally that a degenerate one-dimensional Bose gas relaxes to a state that can be described by such a generalized ensemble. This is verified through a detailed study of correlation functions up to 10th order. The applicability of the generalized ensemble description for isolated quantum many-body systems points to a natural emergence of classical statistical properties from the microscopic unitary quantum evolution.
Copyright © 2015, American Association for the Advancement of Science.

Kibble-Zurek theory models the dynamics of spontaneous symmetry breaking, which plays an important role in a wide variety
of physical contexts, ranging from cosmology to superconductors. We explored these dynamics in a homogeneous system by thermally
quenching an atomic gas with short-range interactions through the Bose-Einstein phase transition. Using homodyne matter-wave
interferometry to measure first-order correlation functions, we verified the central quantitative prediction of the Kibble-Zurek
theory, namely the homogeneous-system power-law scaling of the coherence length with the quench rate. Moreover, we directly
confirmed its underlying hypothesis, the freezing of the correlation length near the transition. Our measurements agree with
a beyond-mean-field theory and support the expectation that the dynamical critical exponent for this universality class is
.

Prethermalization
When a physical system is subjected to a rapid change of conditions (for example, a gas of atoms is allowed to occupy a volume twice the size of the original container), it quickly achieves a new temperature (thermalizes) through collisions. However, in some quantum systems many conserved variables inhibit thermalization; understanding the phases the systems go through in the slowing process is of great interest to cosmologists and physicists. Gring et al. (p. 1318 , published online 31 August) separate an ultracold one-dimensional gas of bosonic atoms into two nearly identical halves, and follow how local differences in phase between the halves evolve in time by examining their interference. Initially, the local phases are almost identical, but a rapid decoherence ensues, followed by a very slow further decay. The authors analyze the relative state reached after the initial fast decay and find that it can be described by an equilibrium function with an effective temperature several times less than the initial temperature. Because this cannot be the final state of the system, the authors term the initial process prethermalization.

This is the first part of what will be a two-part review of distribution functions in physics. Here we deal with fundamentals and the second part will deal with applications. We discuss in detail the properties of the distribution function defined earlier by one of us (EPW) and we derive some new results. Next, we treat various other distribution functions. Among the latter we emphasize the so-called P distribution, as well as the generalized P distribution, because of their importance in quantum optics.

We present a novel imaging system for ultracold quantum gases in expansion. After release from a confining potential, atoms fall through a sheet of resonant excitation laser light and the emitted fluorescence photons are imaged onto an amplified CCD camera using a high numerical aperture optical system. The imaging system reaches an extraordinary dynamic range, not attainable with conventional absorption imaging. We demonstrate single-atom detection for dilute atomic clouds with high efficiency where at the same time dense Bose-Einstein condensates can be imaged without saturation or distortion. The spatial resolution can reach the sampling limit as given by the 8 \mu m pixel size in object space. Pulsed operation of the detector allows for slice images, a first step toward a 3D tomography of the measured object. The scheme can easily be implemented for any atomic species and all optical components are situated outside the vacuum system. As a first application we perform thermometry on rubidium Bose-Einstein condensates created on an atom chip. Comment: 24 pages, 10 figures. v2: as published

The reliable detection of single quantum particles has revolutionized the field of quantum optics and quantum information processing. For several years, researchers have aspired to extend such detection possibilities to larger-scale, strongly correlated quantum systems in order to record in situ images of a quantum fluid in which each underlying quantum particle is detected. Here we report fluorescence imaging of strongly interacting bosonic Mott insulators in an optical lattice with single-atom and single-site resolution. From our images, we fully reconstruct the atom distribution on the lattice and identify individual excitations with high fidelity. A comparison of the radial density and variance distributions with theory provides a precise in situ temperature and entropy measurement from single images. We observe Mott-insulating plateaus with near-zero entropy and clearly resolve the high-entropy rings separating them, even though their width is of the order of just a single lattice site. Furthermore, we show how a Mott insulator melts with increasing temperature, owing to a proliferation of local defects. The ability to resolve individual lattice sites directly opens up new avenues for the manipulation, analysis and applications of strongly interacting quantum gases on a lattice. For example, one could introduce local perturbations or access regions of high entropy, a crucial requirement for the implementation of novel cooling schemes.

Quantum many-body systems can have phase transitions even at zero temperature; fluctuations arising from Heisenberg's uncertainty principle, as opposed to thermal effects, drive the system from one phase to another. Typically, during the transition the relative strength of two competing terms in the system's Hamiltonian changes across a finite critical value. A well-known example is the Mott-Hubbard quantum phase transition from a superfluid to an insulating phase, which has been observed for weakly interacting bosonic atomic gases. However, for strongly interacting quantum systems confined to lower-dimensional geometry, a novel type of quantum phase transition may be induced and driven by an arbitrarily weak perturbation to the Hamiltonian. Here we observe such an effect--the sine-Gordon quantum phase transition from a superfluid Luttinger liquid to a Mott insulator--in a one-dimensional quantum gas of bosonic caesium atoms with tunable interactions. For sufficiently strong interactions, the transition is induced by adding an arbitrarily weak optical lattice commensurate with the atomic granularity, which leads to immediate pinning of the atoms. We map out the phase diagram and find that our measurements in the strongly interacting regime agree well with a quantum field description based on the exactly solvable sine-Gordon model. We trace the phase boundary all the way to the weakly interacting regime, where we find good agreement with the predictions of the one-dimensional Bose-Hubbard model. Our results open up the experimental study of quantum phase transitions, criticality and transport phenomena beyond Hubbard-type models in the context of ultracold gases.

Matter-wave interference experiments enable us to study matter at its most basic, quantum level and form the basis of high-precision sensors for applications such as inertial and gravitational field sensing. Success in both of these pursuits requires the development of atom-optical elements that can manipulate matter waves at the same time as preserving their coherence and phase. Here, we present an integrated interferometer based on a simple, coherent matter-wave beam splitter constructed on an atom chip. Through the use of radio-frequency-induced adiabatic double-well potentials, we demonstrate the splitting of Bose-Einstein condensates into two clouds separated by distances ranging from 3 to 80 microns, enabling access to both tunnelling and isolated regimes. Moreover, by analysing the interference patterns formed by combining two clouds of ultracold atoms originating from a single condensate, we measure the deterministic phase evolution throughout the splitting process. We show that we can control the relative phase between the two fully separated samples and that our beam splitter is phase-preserving.

This paper reviews recent experimental and theoretical progress concerning many-body phenomena in dilute, ultracold gases. It focuses on effects beyond standard weak-coupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, or lowest-Landau-level physics in quasi-two-dimensional gases in fast rotation. Strong correlations in fermionic gases are discussed in optical lattices or near-Feshbach resonances in the BCS-BEC crossover.

We use quantum sine-Gordon model to describe the low energy dynamics of a pair of coupled one-dimensional condensates of interacting atoms. We show that the nontrivial excitation spectrum of the quantum sine-Gordon model, which includes soliton and breather excitations, can be observed in experiments with time-dependent modulation of the tunneling amplitude, potential difference between condensates, or phase of tunneling amplitude. We use the form-factor approach to compute structure factors corresponding to all three types of perturbations.

A quantum breakdown
At low temperatures, two-dimensional (2D) systems with contact interactions are expected to exhibit quantum anomalies—a breakdown of scaling laws that characterize such systems in the classical regime. Signatures of these anomalies have been observed in the real-space properties of 2D Fermi gases, but the effect is much less pronounced than expected on theoretical grounds. Murthy et al. studied the momentum-space profiles of 2D superfluids of fermionic atoms. They initially perturbed the gas and then monitored the momentum distribution of its atoms. In the regime of strong interactions between the atoms, the momentum profiles deviated markedly from the classical scaling.
Science , this issue p. 268

It is difficult to carry out and verify digital quantum simulations that use many quantum bits. A hybrid device based on a digital classical computer and an analog quantum processor suggests a way forward. A programmable analog quantum simulator.

We develop a highly efficient method to numerically simulate thermal fluctuations and correlations in non-relativistic continuous bosonic one-dimensional systems. We start by noticing the equivalence of their description through the transfer matrix formalism and a Fokker-Planck equation for a distribution evolving in space. The corresponding stochastic differential (It\={o}) equation is very suitable for computer simulations, allowing the calculation of arbitrary correlation functions. As an illustration, we apply our method to the case of two tunnel-coupled quasicondensates of bosonic atoms.

In the modern understanding of turbulence, a central concept is the existence of cascades of excitations from large to small lengthscales, or vice-versa. This concept was introduced in 1941 by Kolmogorov and Obukhov, and the phenomenon has since been observed in a variety of systems, including interplanetary plasmas, supernovae, ocean waves, and financial markets. Despite a lot of progress, quantitative understanding of turbulence remains a challenge due to the interplay of many lengthscales that usually thwarts theoretical simulations of realistic experimental conditions. Here we observe the emergence of a turbulent cascade in a weakly interacting homogeneous Bose gas, a quantum fluid that is amenable to a theoretical description on all relevant lengthscales. We prepare a Bose-Einstein condensate (BEC) in an optical box, drive it out of equilibrium with an oscillating force that pumps energy into the system at the largest lengthscale, study the BEC's nonlinear response to the periodic drive, and observe a gradual development of a cascade characterised by an isotropic power-law distribution in momentum space. We numerically model our experiments using the Gross-Pitaevskii equation (GPE) and find excellent agreement with the measurements. Our experiments establish the uniform Bose gas as a promising new platform for investigating many aspects of turbulence, including the interplay of vortex and wave turbulence and the relative importance of quantum and classical effects.

Predicting the dynamics of many-body systems far from equilibrium is a challenging theoretical problem. A long-predicted phenomenon
in hydrodynamic nonequilibrium systems is the occurrence of Sakharov oscillations, which manifest in the anisotropy of the
cosmic microwave background and the large-scale correlations of galaxies. Here, we report the observation of Sakharov oscillations
in the density fluctuations of a quenched atomic superfluid through a systematic study in both space and time domains and
with tunable interaction strengths. Our work suggests a different approach to the study of nonequilibrium dynamics of quantum
many-body systems and the exploration of their analogs in cosmology and astrophysics.

The problem of expanding a density operator rho in forms that simplify the evaluation of important classes of quantum-mechanical expectation values is studied. The weight function P(alpha) of the P representation, the Wigner distribution W(alpha), and the function , where |alpha> is a coherent state, are discussed from a unified point of view. Each of these quasiprobability distributions is examined as the expectation value of a Hermitian operator, as the weight function of an integral representation for the density operator and as the function associated with the density operator by one of the operator-function correspondences defined in the preceding paper. The weight function P(alpha) of the P representation is shown to be the expectation value of a Hermitian operator all of whose eigenvalues are infinite. The existence of the function P(alpha) as an infinitely differentiable function is found to be equivalent to the existence of a well-defined antinormally ordered series expansion for the density operator in powers of the annihilation and creation operators a and a†. The Wigner distribution W(alpha) is shown to be a continuous, uniformly bounded, square-integrable weight function for an integral expansion of the density operator and to be the function associated with the symmetrically ordered power-series expansion of the density operator. The function , which is infinitely differentiable, corresponds to the normally ordered form of the density operator. Its use as a weight function in an integral expansion of the density operator is shown to involve singularities that are closely related to those which occur in the P representation. A parametrized integral expansion of the density operator is introduced in which the weight function W(alpha,s) may be identified with the weight function P(alpha) of the P representation, with the Wigner distribution W(alpha), and with the function when the order parameter s assumes the values s=+1, 0, -1, respectively. The function W(alpha,s) is shown to be the expectation value of the ordered operator analog of the delta function defined in the preceding paper. This operator is in the trace class for Res

Available for the first time in paperback, The Quantum Theory of Fields is a self-contained, comprehensive, and up-to-date introduction to quantum field theory from Nobel Laureate Steven Weinberg. Volume I introduces the foundations of quantum field theory. The development is fresh and logical throughout, with each step carefully motivated by what has gone before. After a brief historical outline, the book begins with the principles of relativity and quantum mechanics, and the properties of particles that follow. Quantum field theory emerges from this as a natural consequence. The classic calculations of quantum electrodynamics are presented in a thoroughly modern way, showing the use of path integrals and dimensional regularization. It contains much original material, and is peppered with examples and insights drawn from the author's experience as a leader of elementary particle research. Exercises are included at the end of each chapter.

The Kadanoff theory of scaling near the critical point for an Ising ferromagnet is cast in differential form. The resulting differential equations are an example of the differential equations of the renormalization group. It is shown that the Widom-Kadanoff scaling laws arise naturally from these differential equations if the coefficients in the equations are analytic at the critical point. A generalization of the Kadanoff scaling picture involving an "irrelevant" variable is considered; in this case the scaling laws result from the renormalization-group equations only if the solution of the equations goes asymptotically to a fixed point.

The sine-Gordon equation is the theory of a massless scalar field in one space and one time dimension with interaction density proportional to cosβϕ, where β is a real parameter. I show that if β2 exceeds 8π, the energy density of the theory is unbounded below; if β2 equals 4π, the theory is equivalent to the zero-charge sector of the theory of a free massive Fermi field; for other values of β, the theory is equivalent to the zero-charge sector of the massive Thirring model. The sine-Gordon soliton is identified with the fundamental fermion of the Thirring model.

Operators for the creation and annihilation of quantum sine-Gordon
solitons are constructed. The operators satisfy the anticommutation
relations and field equations of the massive Thirring model. The results
of Coleman are thus reestablished without the use of perturbation
theory. It is hoped that the method is more generally applicable to a
quantum-mechanical treatment of extended solutions of field theories.

We discuss a phase space representation of quantum dynamics of systems with many degrees of freedom. This representation is based on a perturbative expansion in quantum fluctuations around one of the classical limits. We explicitly analyze expansions around three such limits: (i) corpuscular or Newtonian limit in the coordinate–momentum representation, (ii) wave or Gross–Pitaevskii limit for interacting bosons in the coherent state representation, and (iii) Bloch limit for the spin systems. We discuss both the semiclassical (truncated Wigner) approximation and further quantum corrections appearing in the form of either stochastic quantum jumps along the classical trajectories or the nonlinear response to such jumps. We also discuss how quantum jumps naturally emerge in the analysis of non-equal time correlation functions. This representation of quantum dynamics is closely related to the phase space methods based on the Wigner–Weyl quantization and to the Keldysh technique. We show how such concepts as the Wigner function, Weyl symbol, Moyal product, Bopp operators, and others automatically emerge from the Feynmann's path integral representation of the evolution in the Heisenberg representation. We illustrate the applicability of this expansion with various examples mostly in the context of cold atom systems including sine-Gordon model, one- and two-dimensional Bose–Hubbard model, Dicke model and others.

This paper describes the quantum theory of solitons — the localized solutions of the classical field equations. The scattering matrix for the processes with solitons is defined within the functional integral formalism. The Lorentz-invariant perturbation theory for solitons is consistently set up. The physical properties of solitons are calculated for two-dimensional scalar theories in the one-loop approximation.

It is possible for a classical field theory to have two homogeneous stable equilibrium states with different energy densities. In the quantum version of the theory, the state of higher-energy density becomes unstable through barrier penetration; it is false vacuum. This is a the first of two papers developing the qualitative and quantitative semiclassical theory of the decay of such a false vacuum for theories of a single scalar field with nonderivative interactions. In the limit of vanishing energy density between the two ground states, it is possible to obtain explicit expressions for the relevant quantities to leading order in h; in the more general case, the problem can be reduced to solving a single nonlinear ordinary differential equation.

Recent years have seen tremendous progress in creating complex atomic many-body quantum systems. One approach is to use macroscopic, effectively thermodynamic ensembles of ultracold atoms to create quantum gases and strongly correlated states of matter, and to analyse the bulk properties of the ensemble. For example, bosonic and fermionic atoms in a Hubbard-regime optical lattice can be used for quantum simulations of solid-state models. The opposite approach is to build up microscopic quantum systems atom-by-atom, with complete control over all degrees of freedom. The atoms or ions act as qubits and allow the realization of quantum gates, with the goal of creating highly controllable quantum information systems. Until now, the macroscopic and microscopic strategies have been fairly disconnected. Here we present a quantum gas 'microscope' that bridges the two approaches, realizing a system in which atoms of a macroscopic ensemble are detected individually and a complete set of degrees of freedom for each of them is determined through preparation and measurement. By implementing a high-resolution optical imaging system, single atoms are detected with near-unity fidelity on individual sites of a Hubbard-regime optical lattice. The lattice itself is generated by projecting a holographic mask through the imaging system. It has an arbitrary geometry, chosen to support both strong tunnel coupling between lattice sites and strong on-site confinement. Our approach can be used to directly detect strongly correlated states of matter; in the context of condensed matter simulation, this corresponds to the detection of individual electrons in the simulated crystal. Also, the quantum gas microscope may enable addressing and read-out of large-scale quantum information systems based on ultracold atoms.

We consider the time evolution of nonequilibrium quantum scalar fields in the O(N) model, using the next-to-leading order 1/N expansion of the two-particle irreducible effective action. A comparison with exact numerical simulations in 1+1 dimensions in the classical limit shows that the 1/N expansion gives quantitatively precise results already for moderate values of N. For sufficiently high initial occupation numbers the time evolution of quantum fields is shown to be accurately described by classical physics. Eventually the correspondence breaks down due to the difference between classical and quantum thermal equilibrium.

The temporal development of quantized fields, in its particle aspect, is described by propagation functions, or Green’s functions. The construction of these functions for coupled fields is usually considered from the viewpoint of perturbation theory. Although the latter may be resorted to for detailed calculations, it is desirable to avoid founding the formal theory of the Green’s functions on the restricted basis provided by the assumption of expandability in powers of coupling constants. These notes are a preliminary account of a general theory of Green’s functions, in which the defining property is taken to be the representation of the fields of prescribed sources.

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