5,435 research outputs found
Quantum-limited metrology with product states
We study the performance of initial product states of n-body systems in
generalized quantum metrology protocols that involve estimating an unknown
coupling constant in a nonlinear k-body (k << n) Hamiltonian. We obtain the
theoretical lower bound on the uncertainty in the estimate of the parameter.
For arbitrary initial states, the lower bound scales as 1/n^k, and for initial
product states, it scales as 1/n^(k-1/2). We show that the latter scaling can
be achieved using simple, separable measurements. We analyze in detail the case
of a quadratic Hamiltonian (k = 2), implementable with Bose-Einstein
condensates. We formulate a simple model, based on the evolution of
angular-momentum coherent states, which explains the O(n^(-3/2)) scaling for k
= 2; the model shows that the entanglement generated by the quadratic
Hamiltonian does not play a role in the enhanced sensitivity scaling. We show
that phase decoherence does not affect the O(n^(-3/2)) sensitivity scaling for
initial product states.Comment: 15 pages, 6 figure
Nonlinear Lattice Waves in Random Potentials
Localization of waves by disorder is a fundamental physical problem
encompassing a diverse spectrum of theoretical, experimental and numerical
studies in the context of metal-insulator transition, quantum Hall effect,
light propagation in photonic crystals, and dynamics of ultra-cold atoms in
optical arrays. Large intensity light can induce nonlinear response, ultracold
atomic gases can be tuned into an interacting regime, which leads again to
nonlinear wave equations on a mean field level. The interplay between disorder
and nonlinearity, their localizing and delocalizing effects is currently an
intriguing and challenging issue in the field. We will discuss recent advances
in the dynamics of nonlinear lattice waves in random potentials. In the absence
of nonlinear terms in the wave equations, Anderson localization is leading to a
halt of wave packet spreading.
Nonlinearity couples localized eigenstates and, potentially, enables
spreading and destruction of Anderson localization due to nonintegrability,
chaos and decoherence. The spreading process is characterized by universal
subdiffusive laws due to nonlinear diffusion. We review extensive computational
studies for one- and two-dimensional systems with tunable nonlinearity power.
We also briefly discuss extensions to other cases where the linear wave
equation features localization: Aubry-Andre localization with quasiperiodic
potentials, Wannier-Stark localization with dc fields, and dynamical
localization in momentum space with kicked rotors.Comment: 45 pages, 19 figure
Parallel Self-Consistent-Field Calculations via Chebyshev-Filtered Subspace Acceleration
Solving the Kohn-Sham eigenvalue problem constitutes the most computationally
expensive part in self-consistent density functional theory (DFT) calculations.
In a previous paper, we have proposed a nonlinear Chebyshev-filtered subspace
iteration method, which avoids computing explicit eigenvectors except at the
first SCF iteration. The method may be viewed as an approach to solve the
original nonlinear Kohn-Sham equation by a nonlinear subspace iteration
technique, without emphasizing the intermediate linearized Kohn-Sham eigenvalue
problem. It reaches self-consistency within a similar number of SCF iterations
as eigensolver-based approaches. However, replacing the standard
diagonalization at each SCF iteration by a Chebyshev subspace filtering step
results in a significant speedup over methods based on standard
diagonalization. Here, we discuss an approach for implementing this method in
multi-processor, parallel environment. Numerical results are presented to show
that the method enables to perform a class of highly challenging DFT
calculations that were not feasible before
Quasi-periodic motions in dynamical systems. Review of a renormalisation group approach
Power series expansions naturally arise whenever solutions of ordinary
differential equations are studied in the regime of perturbation theory. In the
case of quasi-periodic solutions the issue of convergence of the series is
plagued of the so-called small divisor problem. In this paper we review a
method recently introduced to deal with such a problem, based on
renormalisation group ideas and multiscale techniques. Applications to both
quasi-integrable Hamiltonian systems (KAM theory) and non-Hamiltonian
dissipative systems are discussed. The method is also suited to situations in
which the perturbation series diverges and a resummation procedure can be
envisaged, leading to a solution which is not analytic in the perturbation
parameter: we consider explicitly examples of solutions which are only
infinitely differentiable in the perturbation parameter, or even defined on a
Cantor set.Comment: 36 pages, 8 figures, review articl
Renormalization Group and the Melnikov Problem for PDE's
We give a new proof of persistence of quasi-periodic, low dimensional
elliptic tori in infinite dimensional systems. The proof is based on a
renormalization group iteration that was developed recently in [BGK] to address
the standard KAM problem, namely, persistence of invariant tori of maximal
dimension in finite dimensional, near integrable systems. Our result covers
situations in which the so called normal frequencies are multiple. In
particular, it provides a new proof of the existence of small-amplitude,
quasi-periodic solutions of nonlinear wave equations with periodic boundary
conditions.Comment: 44 pages, plain Te
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