A phase transition indicates a sudden change in the properties of a large system. For temperaturedriven phase transitions this is related to non-analytic behavior of the free energy density at the critical temperature: The knowledge of the free energy density in one phase is insufficient to predict the properties of the other phase. In this paper we show that a close analogue of this behavior can occur in the real time evolution of quantum systems, namely non-analytic behavior at a critical time. We denote such behavior a dynamical phase transition and explore its properties in the transverse field Ising model. Specifically, we show that the equilibrium quantum phase transition and the dynamical phase transition in this model are intimately related.

When a system thermalizes it loses all local memory of its initial conditions. This is a general feature of open systems and is well described by equilibrium statistical mechanics. Even within a closed (or reversible) quantum system, where unitary time evolution retains all information about its initial state, subsystems can still thermalize using the rest of the system as an effective heat bath. Exceptions to quantum thermalization have been predicted and observed, but typically require inherent symmetries [1,2] or noninteracting particles in the presence of static disorder [3][4][5][6]. The prediction of many-body localization (MBL), in which disordered quantum systems can fail to thermalize in spite of strong interactions and high excitation energy, was therefore surprising and has attracted considerable theoretical attention [3, 7-10]. Here we experimentally generate MBL states by applying an Ising Hamiltonian with long-range interactions and programmably random disorder to ten spins initialized far from equilibrium. We observe the essential signatures of MBL: memory retention of the initial state, a Poissonian distribution of energy level spacings, and entanglement growth in the system at long times. Our platform can be scaled to higher numbers of spins, where detailed modeling of MBL becomes impossible due to the complexity of representing such entangled quantum states. Moreover, the high degree of control in our experiment may guide the use of MBL states as potential quantum memories in naturally disordered quantum systems [11]. i (γ = x, y, z) as well as arbitrary spin correlation functions along any direction.J i,j is a tunable, long-range coupling that falls off approximately algebraically as J i,j ∝ J max /|i − j| α [17], where J max is typically 2π(0.5 kHz). Here we tune α arXiv:1508.07026v1 [quant-ph]

Gauge theories are fundamental to our understanding of interactions between the elementary constituents of matter as mediated by gauge bosons [1,2]. However, computing the real-time dynamics in gauge theories is a notorious challenge for classical computational methods. In the spirit of Feynman's vision of a quantum simulator [3,4], this has recently stimulated theoretical effort to devise schemes for simulating such theories on engineered quantum-mechanical devices, with the difficulty that gauge invariance and the associated local conservation laws (Gauss laws) need to be implemented [5][6][7]. Here we report the first experimental demonstration of a digital quantum simulation of a lattice gauge theory, by realising 1+1-dimensional quantum electrodynamics (Schwinger model [8,9]) on a few-qubit trapped-ion quantum computer. We are interested in the real-time evolution of the Schwinger mechanism [10,11], describing the instability of the bare vacuum due to quantum fluctuations, which manifests itself in the spontaneous creation of electron-positron pairs. To make efficient use of our quantum resources, we map the original problem to a spin model by eliminating the gauge fields [12] in favour of exotic longrange interactions, which have a direct and efficient implementation on an ion trap architecture [13]. We explore the Schwinger mechanism of particle-antiparticle generation by monitoring the mass production and the vacuum persistence amplitude. Moreover, we track the real-time evolution of entanglement in the system, which illustrates how particle creation and entanglement generation are directly related. Our work represents a first step towards quantum simulating high-energy theories with atomic physics experiments, the long-term vision being the extension to real-time quantum simulations of non-Abelian lattice gauge theories.Small-scale quantum computers exist today in the laboratory as programmable quantum devices [14]. In particular, trapped-ion quantum computers [13] provide a platform allowing a few hundred coherent quantum gates on a few qubits, with a clear roadmap towards scaling up FIG. 1. (a)The instability of the vacuum due to quantum fluctuations is one of the most fundamental effects in gauge theories. We simulate the coherent real time dynamics of particle-antiparticle creation by realising the Schwinger model (one-dimensional quantum electrodynamics) on a lattice, as described in the main text. (b) The experimental setup for the simulation consists of a linear Paul trap, where a string of 40 Ca + ions is confined. The electronic states of each ion encode a spin |↑ or |↓ ; these can be manipulated using laser beams (see Methods for details).these devices [4,15]. This provides the tools for universal digital quantum simulation [16], where the time evolution of a quantum system is approximated as a stroboscopic sequence of quantum gates [17]. Here we show how this quantum technology can be used to simulate the real time dynamics of a minimal model of a lattice gauge theory, realising the Schwinge...

Quantum theory provides an extensive framework for the description of the equilibrium properties of quantum matter. Yet experiments in quantum simulators have now opened up a route towards the generation of quantum states beyond this equilibrium paradigm. While these states promise to show properties not constrained by equilibrium principles, such as the equal a priori probability of the microcanonical ensemble, identifying the general properties of nonequilibrium quantum dynamics remains a major challenge, especially in view of the lack of conventional concepts such as free energies. The theory of dynamical quantum phase transitions attempts to identify such general principles by lifting the concept of phase transitions to coherent quantum real-time evolution. This review provides a pedagogical introduction to this field. Starting from the general setting of nonequilibrium dynamics in closed quantum many-body systems, we give the definition of dynamical quantum phase transitions as phase transitions in time with physical quantities becoming nonanalytic at critical times. We summarize the achieved theoretical advances as well as the first experimental observations, and furthermore provide an outlook to major open questions as well as future directions of research.

Dynamical quantum phase transitions (DQPTs) extend the concept of phase transitions and thus universality to the non-equilibrium regime. In this letter, we investigate DQPTs in a string of ions simulating interacting transverse-field Ising models. We observe non-equilibrium dynamics induced by a quantum quench and show for strings of up to 10 ions the direct detection of DQPTs by measuring a quantity that becomes non-analytic in time in the thermodynamic limit. Moreover, we provide a link between DQPTs and the dynamics of other relevant quantities such as the magnetization, and we establish a connection between DQPTs and entanglement production.Today, the equilibrium properties of quantum matter are theoretically described with remarkable success. Yet, in recent years pioneering experiments have created novel quantum states beyond this equilibrium paradigm [1,2]. Thanks to this progress, it is now possible to experimentally study exotic phenomena such as many-body localization [3,4], prethermalization [5, 6], particle-antiparticle production in the lattice Schwinger model [7], and light-induced superconductivity [8]. Understanding general properties of such nonequilibrium quantum states provides a significant challenge, calling for new concepts that extend important principles such as universality to the non-equilibrium realm. A general approach towards this major goal is the theory of dynamical quantum phase transitions (DQPTs) [9], which extends the concept of phase transitions and thus universality to the nonequilibrium regime. In this letter, we directly observe the defining real-time non-analyticities at DQPTs in a trappedion quantum simulator for interacting transverse-field Ising models. Moreover, we provide a link between DQPTs and the dynamics of other relevant quantities such as the magnetization, and we establish a connection between DQPTs and entanglement production. Our work advances towards experimentally characterizing nonequilibrium quantum states and their dynamics, by offering general experimental tools that can be applied also to other inherently dynamical phenomena.Statistical mechanics and thermodynamics provide us with an excellent understanding of equilibrium quantum manybody systems. A key concept in this framework is the canonical partition function Z(T ) = Tr(e −H/k B T ), with T the temperature, k B the Boltzmann constant, and H the system Hamiltonian. The partition function encodes thermodynamics via the free-energy density f = −(k B T/N) log [Z(T )], where N de- * Present address:ARC Centre of Excellence for Engineered Quantum Systems, School of Physics, University of Sydney, NSW, 2006, Australia notes the number of degrees of freedom. A phase transition, i.e., a sudden change of macroscopic behaviour, is associated with a non-analytical behaviour of f as a function of temperature or another control parameter g such as an external magnetic field. Quantum phase transitions (QPTs) [10] occur when T is kept at absolute zero, where the system's groundstate properties undergo a non-analyt...

Recently, the identification of non-equilibrium signatures of topology in the dynamics of such systems has attracted particular attention [3][4][5][6] . Here, we experimentally study the dynamical evolution of the wavefunction using time-and momentum-resolved full state tomography for spin-polarized fermionic atoms in driven optical lattices 7 . We observe the appearance, movement and annihilation of dynamical vortices in momentum space after sudden quenches close to the topological phase transition. These dynamical vortices can be interpreted as dynamical Fisher zeros of the Loschmidt amplitude 8 , which signal a so-called dynamical phase transition 9,10 . Our results pave the way to a deeper understanding of the connection between topological phases and non-equilibrium dynamics.The discovery of topological matter has revolutionized our understanding of band theory: not only are the dispersions of the energy bands important, but so is the geometry of the corresponding eigenstates 1 . The non-local nature of the topological invariants characterizing such phases goes beyond the Landau paradigm of local order parameters and leads to topological protection, for example, against disorder. Ultracold quantum gases in optical lattices allow for controlled studies of archetypal topological models [11][12][13][14] . In addition, compared with, for example condensed-matter systems, they also allow for detailed studies of the relation between dynamics and topology as the timescales are experimentally easier to access. Dynamical studies of driven systems have recently attracted attention in terms of their high T c superconductivity 15 . A particular challenge is to identify non-equilibrium signatures of topology in the dynamics of highly excited states 3,4,16 . Here, we observe the time evolution of the wavefunction after a sudden quench in a Haldanelike model and find dynamical vortices as a signature of the topological nature of the underlying ground state.In the experiments described here, the state tomography method allows mapping of the full quantum-mechanical wavefunction of non-interacting ultracold fermionic quantum gases in an optical lattice for any time after a sudden quench of the system close to or into a Chern insulating phase. As a key result, we identify in an intense series of measurements the appearance, movement and annihilation In the initial system, tunnelling J AB between the A and B sites is suppressed by a large energy offset. In the final Floquet system, tunnelling is re-established by means of near-resonant driving. b, At each momentum, the Hamiltonian describes the coupling between the states of the A and B sublattices, and can be visualized on a Bloch sphere. In the initial system, the Hamiltonian for all momenta points to the north pole, whereas in the Floquet system, the Hamiltonian covers a large surface of the Bloch sphere. c, Phase diagram for the Floquet Hamiltonian as a function of shaking amplitude and detuning with respect to the sublattice offset for the case of circular lattice shaking...

Entanglement plays a central role in our understanding of quantum many body physics, and is fundamental in characterising quantum phases and quantum phase transitions. Developing protocols to detect and quantify entanglement of many-particle quantum states is thus a key challenge for present experiments. Here, we show that the quantum Fisher information, representing a witness for genuinely multipartite entanglement, becomes measurable for thermal ensembles via the dynamic susceptibility, i.e., with resources readily available in present cold atomic gas and condensed-matter experiments. This moreover establishes a fundamental connection between multipartite entanglement and many-body correlations contained in response functions, with profound implications close to quantum phase transitions. There, the quantum Fisher information becomes universal, allowing us to identify strongly entangled phase transitions with a divergent multipartiteness of entanglement. We illustrate our framework using paradigmatic quantum Ising models, and point out potential signatures in optical-lattice experiments.Entanglement is a central theoretical concept underlying the characterisation of quantum many-body states in condensed-matter and high-energy physics, as well as quantum information. For example, entanglement properties reveal exotic states of matter such as topological spin liquids [1] or many-body localization [2,3], the holographic entanglement entropy identifies confinement/deconfinement transitions in gauge theories [4,5], and entanglement is considered the central resource for quantum-enhanced metrology [6,7] as well as quantum computation [8][9][10][11]. In experiments, entanglement becomes measurable via a tomographic determination of the many-particle quantum state [12][13][14][15], and protocols have been developed [16] and implemented in remarkable experiments [17] to measure entanglement entropies in quench dynamics and quantum phase transitions. However, the resources required by these protocols scale exponentially with the system size, and these experimental efforts are thus limited a priori to few-particle systems.To address the problem of detecting and quantifying multipartite entanglement for large systems, we consider below the quantum Fisher information (QFI) as an entanglement witness [18][19][20]. Our key result is thatfor a many-body system at thermal equilibrium at any temperature-the QFI can be determined directly from a measurement of Kubo linear response functions, in particular the dynamic susceptibility (see Fig. 1). We emphasise that this measurement prescription is independent of microscopic details of the system of interest and that the measurement of linear response is a standard tool in experiments. Importantly, only modest measurement resources are required that do not scale with system size. The presented prescription therefore makes multipartite entanglement observable for a large variety of * philipp.hauke@uibk.ac.at χ (ω, T ) gives the quantum Fisher information (shaded areas). (c) This pro...

We introduce a topological quantum number -coined dynamical topological order parameter (DTOP) -that is dynamically defined in the real-time evolution of a quantum many-body system and represented by a momentum space winding number of the Pancharatnam geometric phase. Our construction goes conceptually beyond the standard notion of topological invariants characterizing the wave-function of a system, which are constants of motion under coherent time evolution. In particular, we show that the DTOP can change its integer value at discrete times where so called dynamical quantum phase transitions occur, thus serving as a dynamical analog of an order parameter. Interestingly, studying quantum quenches in one-dimensional two-banded Bogoliubov de Gennes models, we find that the DTOP is capable of resolving if the topology of the system Hamiltonian has changed over the quench. Furthermore, we investigate the relation of the DTOP to the dynamics of the string order parameter that characterizes the topology of such systems in thermal equilibrium. arXiv:1504.05599v3 [cond-mat.mes-hall]