For many years, the Luttinger liquid theory has served as a useful paradigm for the description of one-dimensional (1D) quantum fluids in the limit of low energies. This theory is based on a linearization of the dispersion relation of the particles constituting the fluid. We review the recent progress in understanding 1D quantum fluids beyond the low-energy limit, where the nonlinearity of the dispersion relation becomes essential. The novel methods which have been developed to tackle such systems combine phenomenology built on the ideas of the Fermi edge singularity and the Fermi liquid theory, perturbation theory in the interaction strength, and new ways of treating finitesize properties of integrable models. These methods can be applied to a wide variety of 1D fluids, from 1D spin liquids to electrons in quantum wires to cold atoms confined by 1D traps. We review existing results for various dynamic correlation functions, in particular the dynamic structure factor and the spectral function. Moreover, we show how a dispersion nonlinearity leads to finite particle lifetimes, and its impact on the transport properties of 1D systems at finite temperatures is discussed. The conventional Luttinger liquid theory is a special limit of the new theory, and we explain the relation between the two.

We consider insulating phases of cold spin-1 bosonic particles with antiferromagnetic interactions, such as 23 Na, in optical lattices. We show that spin-exchange interactions give rise to several distinct phases, which differ in their spin correlations. In two-and three-dimensional lattices, insulating phases with an odd number of particles per site are always nematic. For insulating states with an even number of particles per site, there is always a spin-singlet phase, and there may also be a first-order transition into the nematic phase. The nematic phase breaks spin rotational symmetry but preserves time reversal symmetry, and has gapless spin-wave excitations. The spin-singlet phase does not break spin symmetry and has a gap to all excitations. In onedimensional lattices, insulating phases with an odd number of particles per site always have a regime where translational symmetry is broken and the ground state is dimerized. We discuss signatures of various phases in Bragg scattering and time-of-flight measurements.

One-dimensional quantum fluids are conventionally described by using an effective hydrodynamic approach known as Luttinger liquid theory. As the principal simplification, a generic spectrum of the constituent particles is replaced by a linear one, which leads to a linear hydrodynamic theory. We show that to describe the measurable dynamic response functions one needs to take into account the nonlinearity of the generic spectrum and thus of the resulting quantum hydrodynamic theory. This nonlinearity leads, for example, to a qualitative change in the behavior of the spectral function. The universal theory developed in this article is applicable to a wide class of one-dimensional fermionic, bosonic, and spin systems.The development of the universal effective description of many-body phenomena is a central problem of the condensed matter theory. The hydrodynamic approach known as Luttinger liquid (LL) theory (1, 2, 3) is routinely applied to one-dimensional (1D) interacting systems. As a crucial simplification, a generic spectrum of the constituent particles is replaced by a linear one, leading to a linear hydrodynamic theory, which is nothing but a collection of noninteracting oscillators. However, to understand a variety of phenomena, such as Coulomb drag between quantum wires (4), momentum-resolved tunneling of electrons in nanowires (5), and neutron scattering off spin chains (3), one needs to take into account the nonlinearity of the spectrum. From classical physics, it is known that the existence of nonlinearities may result in qualitatively new phenomena, such as propagation of solitons and appearance of shock waves. These phenomena take place in a variety of experimental situations because classical nonlinear hydrodynamics is universal: It is phenomenologically derived from simple assumptions, which do not rely on microscopic details. Although description of linear quantum hydrodynamic theory requires only quantum mechanical treatment of noninteracting oscillators, formulation of nonlinear quantum hydrodynamics remains a challenging task because of divergences typical of 1

The recent experimental realization of strongly imbalanced mixtures of ultracold atoms opens new possibilities for studying impurity dynamics in a controlled setting. In this paper, we discuss how the techniques of atomic physics can be used to explore new regimes and manifestations of Anderson's orthogonality catastrophe (OC), which could not be accessed in solid-state systems. Specifically, we consider a system of impurity atoms, localized by a strong optical-lattice potential, immersed in a sea of itinerant Fermi atoms. We point out that the Ramsey-interference-type experiments with the impurity atoms allow one to study the OC in the time domain, while radio-frequency (RF) spectroscopy probes the OC in the frequency domain. The OC in such systems is universal, not only in the long-time limit, but also for all times and is determined fully by the impurity-scattering length and the Fermi wave vector of the itinerant fermions. We calculate the universal Ramsey response and RF-absorption spectra. In addition to the standard power-law contributions, which correspond to the excitation of multiple particle-hole pairs near the Fermi surface, we identify a novel, important contribution to the OC that comes from exciting one extra particle from the bottom of the itinerant band. This contribution gives rise to a nonanalytic feature in the RF-absorption spectra, which shows a nontrivial dependence on the scattering length, and evolves into a true power-law singularity with the universal exponent 1=4 at the unitarity. We extend our discussion to spin-echo-type experiments, and show that they probe more complicated nonequilibirum dynamics of the Fermi gas in processes in which an impurity switches between states with different interaction strength several times; such processes play an important role in the Kondo problem, but remained out of reach in the solid-state systems. We show that, alternatively, the OC can be seen in the energy-counting statistics of the Fermi gas following a sudden quench of the impurity state. The energy distribution function, which can be measured in time-of-flight experiments, exhibits characteristic power-law singularities at low energies. Finally, systems in which the itinerant fermions have two or more hyperfine states provide an even richer playground for studying nonequilibrium impurity physics, allowing one to explore the nonequilibrium OC and even to simulate quantum transport through nanostructures. This provides a previously missing connection between cold atomic systems and mesoscopic quantum transport.

The probabilistic character of the measurement process is one of the most puzzling and fascinating aspects of quantum mechanics. In many-body systems quantum mechanical noise reveals non-local correlations of the underlying many-body states. Here, we provide a complete experimental analysis of the shot-to-shot variations of interference fringe contrast for pairs of independently created one-dimensional Bose condensates. Analyzing different system sizes we observe the crossover from thermal to quantum noise, reflected in a characteristic change in the distribution functions from Poissonian to Gumbel-type, in excellent agreement with theoretical predictions based on the Luttinger liquid formalism. We present the first experimental observation of quasi long-range order in one-dimensional atomic condensates, which is a hallmark of quantum fluctuations in one-dimensional systems. Furthermore, our experiments constitute the first analysis of the full distribution of quantum noise in an interacting many-body system.