3,851 research outputs found
Scaling and Universality of the Complexity of Analog Computation
We apply a probabilistic approach to study the computational complexity of
analog computers which solve linear programming problems. We analyze
numerically various ensembles of linear programming problems and obtain, for
each of these ensembles, the probability distribution functions of certain
quantities which measure the computational complexity, known as the convergence
rate, the barrier and the computation time. We find that in the limit of very
large problems these probability distributions are universal scaling functions.
In other words, the probability distribution function for each of these three
quantities becomes, in the limit of large problem size, a function of a single
scaling variable, which is a certain composition of the quantity in question
and the size of the system. Moreover, various ensembles studied seem to lead
essentially to the same scaling functions, which depend only on the variance of
the ensemble. These results extend analytical and numerical results obtained
recently for the Gaussian ensemble, and support the conjecture that these
scaling functions are universal.Comment: 22 pages, latex, 12 eps fig
A numerical and symbolical approximation of the Nonlinear Anderson Model
A modified perturbation theory in the strength of the nonlinear term is used
to solve the Nonlinear Schroedinger Equation with a random potential. It is
demonstrated that in some cases it is more efficient than other methods.
Moreover we obtain error estimates. This approach can be useful for the
solution of other nonlinear differential equations of physical relevance.Comment: 21 pages and 7 figure
Echoes and revival echoes in systems of anharmonically confined atoms
We study echoes and what we call 'revival echoes' for a collection of atoms
that are described by a single quantum wavefunction and are confined in a
weakly anharmonic trap. The echoes and revival echoes are induced by applying
two, successive temporally localized potential perturbations to the confining
potential, one at time , and a smaller one at time . Pulse-like
responses in the expectation value of position are predicted at $t
\approx n\tau$ ($n=2,3,...$) and are particularly evident at $t \approx 2\tau$.
A novel result of our study is the finding of 'revival echoes'. Revivals (but
not echoes) occur even if the second perturbation is absent. In particular, in
the absence of the second perturbation, the response to the first perturbation
dies away, but then reassembles, producing a response at revival times $mT_x$
($m=1,2,...$). Including the second perturbation at $t=\tau$, we find
temporally localized responses, revival echoes, both before and after $t\approx
mT_x$, e.g., at $t\approx m T_x-n \tau$ (pre-revival echoes) and at $t\approx
mT_x+n\tau$, (post-revival echoes), where $m$ and $n$ are $1,2,...$ . Depending
on the form of the perturbations, the 'principal' revival echoes at $t \approx
T_x \pm \tau$ can be much larger than the echo at $t \approx 2\tau$. We develop
a perturbative model for these phenomena, and compare its predictions to the
numerical solutions of the time-dependent Schr\"odinger Equation. The scaling
of the size of the various echoes and revival echoes as a function of the
symmetry and size of the perturbations applied at $t=0$ and $t=\tau$ is
investigated. We also study the presence of revivals and revival echoes in
higher moments of position, , , and the effect of atom-atom
interactions on these phenomena.Comment: 33 pages, 13 figures, corrected typos and added reference
Decay of Quantum Accelerator Modes
Experimentally observable Quantum Accelerator Modes are used as a test case
for the study of some general aspects of quantum decay from classical stable
islands immersed in a chaotic sea. The modes are shown to correspond to
metastable states, analogous to the Wannier-Stark resonances. Different regimes
of tunneling, marked by different quantitative dependence of the lifetimes on
1/hbar, are identified, depending on the resolution of KAM substructures that
is achieved on the scale of hbar. The theory of Resonance Assisted Tunneling
introduced by Brodier, Schlagheck, and Ullmo [9], is revisited, and found to
well describe decay whenever applicable.Comment: 16 pages, 11 encapsulated postscript figures (figures with a better
resolution are available upon request to the authors); added reference for
section
Experimental observation of high-order quantum accelerator modes.
Using a freely falling cloud of cold cesium atoms periodically kicked by pulses from a vertical standing wave of laser light, we present the first experimental observation of high-order quantum accelerator modes. This confirms the recent prediction by Fishman, Guarneri, and Rebuzzini [Phys. Rev. Lett.10.1103/PhysRevLett.89.084101 89, 084101 (2002)]. We also show how these accelerator modes can be identified with the stable regions of phase space in a classical-like chaotic system, despite their intrinsically quantum origin
The Quantum-Classical Crossover in the Adiabatic Response of Chaotic Systems
The autocorrelation function of the force acting on a slow classical system,
resulting from interaction with a fast quantum system is calculated following
Berry-Robbins and Jarzynski within the leading order correction to the
adiabatic approximation. The time integral of the autocorrelation function is
proportional to the rate of dissipation. The fast quantum system is assumed to
be chaotic in the classical limit for each configuration of the slow system. An
analytic formula is obtained for the finite time integral of the correlation
function, in the framework of random matrix theory (RMT), for a specific
dependence on the adiabatically varying parameter. Extension to a wider class
of RMT models is discussed. For the Gaussian unitary and symplectic ensembles
for long times the time integral of the correlation function vanishes or falls
off as a Gaussian with a characteristic time that is proportional to the
Heisenberg time, depending on the details of the model. The fall off is
inversely proportional to time for the Gaussian orthogonal ensemble. The
correlation function is found to be dominated by the nearest neighbor level
spacings. It was calculated for a variety of nearest neighbor level spacing
distributions, including ones that do not originate from RMT ensembles. The
various approximate formulas obtained are tested numerically in RMT. The
results shed light on the quantum to classical crossover for chaotic systems.
The implications on the possibility to experimentally observe deterministic
friction are discussed.Comment: 26 pages, including 6 figure
Antiresonance and Localization in Quantum Dynamics
The phenomenon of quantum antiresonance (QAR), i.e., exactly periodic
recurrences in quantum dynamics, is studied in a large class of nonintegrable
systems, the modulated kicked rotors (MKRs). It is shown that asymptotic
exponential localization generally occurs for (a scaled ) in the
infinitesimal vicinity of QAR points . The localization length
is determined from the analytical properties of the kicking potential. This
``QAR-localization" is associated in some cases with an integrable limit of the
corresponding classical systems. The MKR dynamical problem is mapped into
pseudorandom tight-binding models, exhibiting dynamical localization (DL). By
considering exactly-solvable cases, numerical evidence is given that
QAR-localization is an excellent approximation to DL sufficiently close to QAR.
The transition from QAR-localization to DL in a semiclassical regime, as
is varied, is studied. It is shown that this transition takes place via a
gradual reduction of the influence of the analyticity of the potential on the
analyticity of the eigenstates, as the level of chaos is increased.Comment: To appear in Physical Review E. 51 pre-print pages + 9 postscript
figure
Extended Standard Map with Spatio-Temporal Asymmetry
We analyze the transport properties of a set of symmetry-breaking extensions
%, both spatial and temporal, of the Chirikov--Taylor Map. The spatial and
temporal asymmetries result in the loss of periodicity in momentum direction in
the phase space dynamics, enabling the asymmetric diffusion which is the origin
of the unidirectional motion. The simplicity of the model makes the calculation
of the global dynamical properties of the system feasible both in phase space
and in controlling-parameter space. We present the results of numerical
experiments which show the intricate dependence of the asymmetric diffusion to
the controlling parameters.Comment: 6 pages latex 2e with 12 epsf fig
Mean Field Theory of Josephson Junction Arrays with Charge Frustration
Using the path integral approach, we provide an explicit derivation of the
equation for the phase boundary for quantum Josephson junction arrays with
offset charges and non-diagonal capacitance matrix. For the model with nearest
neighbor capacitance matrix and uniform offset charge , we determine,
in the low critical temperature expansion, the most relevant contributions to
the equation for the phase boundary. We explicitly construct the charge
distributions on the lattice corresponding to the lowest energies. We find a
reentrant behavior even with a short ranged interaction. A merit of the path
integral approach is that it allows to provide an elegant derivation of the
Ginzburg-Landau free energy for a general model with charge frustration and
non-diagonal capacitance matrix. The partition function factorizes as a product
of a topological term, depending only on a set of integers, and a
non-topological one, which is explicitly evaluated.Comment: LaTex, 24 pages, 8 figure
Spin-glass behaviour on random lattices
The ground-state phase diagram of an Ising spin-glass model on a random graph
with an arbitrary fraction of ferromagnetic interactions is analysed in the
presence of an external field. Using the replica method, and performing an
analysis of stability of the replica-symmetric solution, it is shown that
, correponding to an unbiased spin glass, is a singular point in the
phase diagram, separating a region with a spin-glass phase () from a
region with spin-glass, ferromagnetic, mixed, and paramagnetic phases
()
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