Abstract:
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.

Abstract:
We study shot (counting) noise of the amplitude of interference between independent atomic systems. In particular, for the two interfering systems the variance of the fringe amplitude decreases as the inverse power of the number of particles per system with the coefficient being a non-universal number. This number depends on the details of the initial state of each system so that the shot noise measurements can be used to distinguish between such states. We explicitly evaluate this coefficient for the two cases of the interference between bosons in number states and in broken symmetry states. We generalize our analysis to the interference of multiple independent atomic systems. We show that the variance of the interference contrast vanishes as the inverse power of the number of the interfering systems. This result, implying high signal to noise ratio in the interference experiments, holds both for bosons and for fermions.

Abstract:
We derive a microscopic expression for the instantaneous diagonal elements of the density matrix $\rho_{nn}(t)$ in the adiabatic basis for an arbitrary time dependent process in a closed Hamiltonian system. If the initial density matrix is stationary (diagonal) then this expression contains only squares of absolute values of matrix elements of the evolution operator, which can be interpreted as transition probabilities. We then derive the microscopic expression for the heat defined as the energy generated due to transitions between instantaneous energy levels. If the initial density matrix is passive (diagonal with $\rho_{nn}(0)$ monotonically decreasing with energy) then the heat is non-negative in agreement with basic expectations of thermodynamics. Our findings also can be used for systematic expansion of various observables around the adiabatic limit.

Abstract:
We define a diagonal entropy (d-entropy) for an arbitrary Hamiltonian system as $S_d=-\sum_n \rho_{nn}\ln \rho_{nn}$ with the sum taken over the basis of instantaneous energy states. In equilibrium this entropy coincides with the conventional von Neumann entropy $S_n=-{\rm Tr}\, \rho\ln\rho$. However, in contrast to $S_n$, the d-entropy is not conserved in time in closed Hamiltonian systems. If the system is initially in stationary state then in accord with the second law of thermodynamics the d-entropy can only increase or stay the same. We also show that the d-entropy can be expressed through the energy distribution function and thus it is measurable, at least in principle. Under very generic assumptions of the locality of the Hamiltonian and non-integrability the d-entropy becomes a unique function of the average energy in large systems and automatically satisfies the fundamental thermodynamic relation. This relation reduces to the first law of thermodynamics for quasi-static processes. The d-entropy is also automatically conserved for adiabatic processes. We illustrate our results with explicit examples and show that $S_d$ behaves consistently with expectations from thermodynamics.

Abstract:
We study temporal behavior of a quantum system under a slow external perturbation, which drives the system across a second order quantum phase transition. It is shown that despite the conventional adiabaticity conditions are always violated near the critical point, the number of created excitations still goes to zero in the limit of infinitesimally slow variation of the tuning parameter. It scales with the adiabaticity parameter as a power related to the critical exponents $z$ and $\nu$ characterizing the phase transition. We support general arguments by direct calculations for the Boson Hubbard and the transverse field Ising models.

Abstract:
We develop a consistent perturbation theory in quantum fluctuations around the classical evolution of a system of interacting bosons. The zero order approximation gives the classical Gross-Pitaevskii equations. In the next order we recover the truncated Wigner approximation, where the evolution is still classical but the initial conditions are distributed according to the Wigner transform of the initial density matrix. Further corrections can be characterized as quantum scattering events, which appear in the form of a nonlinear response of the observable to an infinitesimal displacement of the field along its classical evolution. At the end of the paper we give a few numerical examples to test the formalism.

Abstract:
We consider dynamics of boson condensates in finite optical lattices under a slow external perturbation which brings the system to the unstable equilibrium. It is shown that quantum fluctuations drive the condensate into the maximally entangled state. We argue that the truncated Wigner approximation being a natural generalization of the Gross-Pitaevskii classical equations of motion is adequate to correctly describe the time evolution including both collapse and revival of the condensate.

Abstract:
We present theoretical investigation of a single magnetic impurity in a d-wave superconductor using the large N limit. It is shown that the Kondo screening occurs only in the presence of the particle-hole asymmetry. We find analytical expressions for the Kondo temperature, magnetic susceptibility and scattering matrix near the phase transition. The results are generalized for the density of states vanishing with an arbitrary exponent. Also we briefly study the modifications of the theory for the case of a non-magnetic impurity which induces a staggered spin configuration on the nearby copper atoms.

Abstract:
Instanton methods, in which imaginary-time evolution gives the tunneling rate, have been widely used for studying quantum tunneling in various contexts. Nevertheless, how accurate instanton methods are for the problems of macroscopic quantum tunneling (MQT) still remains unclear because of lack of their direct comparison with exact time evolution of the many-body Schroedinger equation. Here, we verify instanton methods applied to coherent MQT. Specifically applying the quasi-exact numerical method of time-evolving block decimation to the system of bosons in a ring lattice, we directly simulate the real-time quantum dynamics of supercurrents, where a coherent oscillation between two macroscopically distinct current states occurs due to MQT. The tunneling rate extracted from the coherent oscillation is compared with that given by the instanton method. We show that the error is within 10% when the effective Planck's constant is sufficiently small. We also discuss phase slip dynamics associated with the coherent oscillations.

Abstract:
We analyze the problem of optimal adiabatic passage through a quantum critical point. We show that to minimize the number of defects the tuning parameter should be changed as a power-law in time. The optimal power is proportional to the logarithm of the total passage time multiplied by universal critical exponents characterizing the phase transition. We support our results by the general scaling analysis and by explicit calculations for the transverse field Ising model.