This Colloquium gives an overview of recent theoretical and experimental progress in the area of nonequilibrium dynamics of isolated quantum systems There is particularly a focus on quantum quenches: the temporal evolution following a sudden or slow change of the coupling constants of the system Hamiltonian Several aspects of the slow dynamics in driven systems are discussed and the universality of such dynamics in gapless systems with specific focus on dynamics near continuous quantum phase transitions is emphasized Recent progress on understanding thermalization in closed systems through the eigenstate thermalization hypothesis is also reviewed and relaxation in integrable systems is discussed Finally key experiments probing quantum dynamics in cold atom systems are overviewed and put into the context of our current theoretical understanding

Colloquium: Nonequilibrium dynamics of closed interacting quantum systems

We review recent developments in the physics of ultracold atomic and molecular gases in optical lattices. Such systems are nearly perfect realisations of various kinds of Hubbard models, and as such may very well serve to mimic condensed matter phenomena. We show how these systems may be employed as quantum simulators to answer some challenging open questions of condensed matter, and even high energy physics. After a short presentation of the models and the methods of treatment of such systems, we discuss in detail, which challenges of condensed matter physics can be addressed with (i) disordered ultracold lattice gases, (ii) frustrated ultracold gases, (iii) spinor lattice gases, (iv) lattice gases in “artificial” magnetic fields, and, last but not least, (v) quantum information processing in lattice gases. For completeness, also some recent progress related to the above topics with trapped cold gases will be discussed. Motto: There are more things in heaven and earth, Horatio, Than are dreamt of in your...

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Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond

What entanglement is present in naturally occurring physical systems at thermal equilibrium? Most such systems are intractable and it is desirable to study simple but realistic systems that can be solved. An example of such a system is the one-dimensional infinite-lattice anisotropic XY model. This model is exactly solvable using the Jordan-Wigner transform, and it is possible to calculate the two-site reduced density matrix for all pairs of sites. Using the two-site density matrix, the entanglement of formation between any two sites is calculated for all parameter values and temperatures. We also study the entanglement in the transverse Ising model, a special case of the XY model, which exhibits a quantum phase transition. It is found that the next-nearest-neighbor entanglement (though not the nearest-neighbor entanglement) is a maximum at the critical point. Furthermore, we show that the critical point in the transverse Ising model corresponds to a transition in the behavior of the entanglement between a single site and the remainder of the lattice.

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Entanglement in a simple quantum phase transition

The development of wave optics for light brought many new insights into our understanding of physics, driven by fundamental experiments like the ones by Young, Fizeau, Michelson-Morley and others. Quantum mechanics, and especially the de Broglie’s postulate relating the momentum p of a particle to the wave vector k of an matter wave: k = 2 λ = p/ℏ, suggested that wave optical experiments should be also possible with massive particles (see table 1), and over the last 40 years electron and neutron interferometers have demonstrated many fundamental aspects of quantum mechanics [1].

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Optics and interferometry with atoms and molecules

We study the unitary time evolution of the entropy of entanglement of a one-dimensional system between the degrees of freedom in an interval of length and its complement, starting from a pure state which is not an eigenstate of the Hamiltonian. We use path integral methods of quantum field theory as well as explicit computations for the transverse Ising spin chain. In both cases, there is a maximum speed v of propagation of signals. In general the entanglement entropy increases linearly with time t up to , after which it saturates at a value proportional to , the coefficient depending on the initial state. This behaviour may be understood as a consequence of causality.

https://iopscience.iop.org/article/10.1088/1742-5468/2005/04/P04010/pdf

Evolution of entanglement entropy in one-dimensional systems

The Liouville equation for the $\mathrm{XY}$ model is solved exactly, and the magnetization is computed explicitly. Nonergodic behavior of the magnetization is found for a general class of time-dependent magnetic fields.

Statistical Mechanics of the XY Model. I

The phase shift caused by rotating a neutron or optical interferometer is derived as the Doppler effect due to the moving source and moving reflecting crystals.

Phase shift in a rotating neutron or optical interferometer

A new general expression for the electrical resistivity of a substance is obtained with the help of projection techniques with the Liouville equation as the point of departure. No many-body detail is sacrificed and the "${\ensuremath{\lambda}}^{2}t$ limit" is not invoked. The first-order result in a perturbation expansion in orders of the scattering is presented in explicit form and shown to have a simple and physical appearance. It is also shown to reduce to the well-known Boltzmann expressions for simple cases.

Theory of Electrical Resistivity

Physical algebras in four dimensions. I. The Clifford algebra in Minkowski spacetime

Assuming the existence of certain limits we give an exact solution of a two-dimensional elastic Ising model with quadratic coupling. The main results are these: (I) If the lattice is slightly stretched, thermodynamic instability occurs for temperatures in a neighborhood of the critical point. (II) Outside this neighborhood, ${\ensuremath{\kappa}}_{T}$, $\ensuremath{\alpha}$, and ${C}_{p}$ diverge. (III) The nearest-neighbor spin-spin correlation function is nonzero for any pressure when $T$ becomes infinite. (IV) Antiferromagnetic as well as ferromagnetic behavior can occur for weak coupling at any pressure. (V) For strong coupling, at 0 pressure the model can have a Curie and N\'eel point or no critical point.

Max Dresden

Papers

Statistical Mechanics of the XY Model. I

Phase shift in a rotating neutron or optical interferometer

Theory of Electrical Resistivity

Physical algebras in four dimensions. I. The Clifford algebra in Minkowski spacetime

Effects of Mechanical Stretching and Quadratic Coupling on Critical Behavior