arXiv subjects

Shmuel Fishman

Publications and source records attributed to Shmuel Fishman.

At least 37 records · Page 2Linked to original sources

arXiv2014

Semiclassical analysis of Bose-Hubbard dynamics

In this work the two site Bose-Hubbard model is studied analytically in the limit of weak coupling u and large number of particles N . The semiclassical approximation where \frac{1}{N} plays the role of Planck's constant was used and perturbation theory to order u^{2} was applied. In particular, the difference in the occupation between the two sites, where initially all particles are at one site was calculated analytically. Excellent agreement with the exact numerical solution was found. This quantity exhibits collapses and revivals that superimpose rapid oscillations. The occupation difference was calculated also for the case where initially both sites are occupied provided that the difference in occupation is sufficiently large. It provides an analytical description of results that were so far found only numerically. Similar behavior and analysis are expected for a large variety of physical situations in optics, atom optics and quantum dynamics of electrons in Rydberg atoms.

quant-ph↗

arXiv2014

Effects of interactions on the dynamics of driven cold atoms

The quantum fidelity was introduced by Peres to study some fingerprints of classically chaotic behavior in the quantum dynamics of the corresponding systems. In the present paper the signatures of classical dynamics near elliptic points and of interactions between particles are characterized for kicked systems. In particular, the period of the fidelity resulting of the interactions is found using analytical and numerical calculations. A mechanism leading to the oscillations with the intermediate period is proposed. It is of a semiclassical origin and results of the interplay between the oscillations of the width of the wave packets and the rotation of their center around the elliptic fixed point.

quant-ph↗

arXiv2014

A simple model for interactions and corrections to the Gross-Pitaevskii Equation

One of the assumptions leading to the Gross-Pitaevskii Equation (GPE) is that the interaction between atom pairs can be written effectively as a δ-function so that the interaction range of the particles is assumed to vanish. A simple model that takes into account the extension of the inter-particle potential is introduced. The correction to the GPE predictions for the energy of a condensate confined by a harmonic trap in the Thomas-Fermi (TF) regime is estimated. Although it is found to be small, we believe that in some situations it can be measured using its dependance on the frequency of the confining trap. Due to the simplicity of the model, it may have a wide range of applications.

cond-mat.quant-gas↗

arXiv2014

From classical to quantum criticality

We study the crossover from classical to quantum phase transitions at zero temperature within the framework of $ϕ^4$ theory. The classical transition at zero temperature can be described by the Landau theory, turning into a quantum Ising transition with the addition of quantum fluctuations. We perform a calculation of the transition line in the regime where the quantum fluctuations are weak. The calculation is based on a renormalization group analysis of the crossover between classical and quantum transitions, and is well controlled even for space-time dimensionality $D$ below 4. In particular, for $D=2$ we obtain an analytic expression for the transition line which is valid for a wide range of parameters, as confirmed by numerical calculations based on the Density Matrix Renormalization Group. This behavior could be tested by measuring the phase diagram of the linear-zigzag instability in systems of trapped ions or repulsively-interacting dipoles.

cond-mat.stat-mech↗

arXiv2013

Multiscale time averaging, Reloaded

We develop a rigorously controlled multi-time scale averaging technique; the averaging is done on a finite time interval, properly chosen, and then, via iterations and normal form transformations, the time intervals are scaled to arbitrary order. Here, we consider as an example the problem of finite dimensional conservative dynamical system, which is quasiperiodic and dominated by slow frequencies, leading to small divisor problems in perturbative schemes.

math-ph↗

arXiv2013

Anderson localisation in laser kicked molecules

The paper explores the prospects of observing the phenomenon of dynamical Anderson localisation via non-resonant Raman-type rotational excitation of molecules by periodic trains of short laser pulses. We define conditions for such an experiment, and show that current femtosecond technology used for non-adiabatic laser alignment of linear molecules is sufficient for this task. Several observables which can serve as indicator for Anderson localisation are suggested for measurement, and the influence of experimental limitations imposed by laser intensity noise, finite pulse duration, limited number of pulses in a train, and thermal effects is analysed.

quant-ph↗

arXiv2012

Dynamic Localization of Interacting Particles in an Anharmonic Potential

We investigate the effect of anharmonicity and interactions on the dynamics of an initially Gaussian wavepacket in a weakly anharmonic potential. We note that depending on the strength and sign of interactions and anharmonicity, the quantum state can be either localized or delocalized in the potential. We formulate a classical model of this phenomenon and compare it to quantum simulations done for a self consistent potential given by the Gross-Pitaevskii Equation.

quant-ph↗

arXiv2012

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 $t=0$, and a smaller one at time $t=τ$. Pulse-like responses in the expectation value of position $ $ are predicted at $t \approx nτ$ ($n=2,3,...$) and are particularly evident at $t \approx 2τ$. 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 $t=τ$, we find temporally localized responses, revival echoes, both before and after $t\approx mT_x$, e.g., at $t\approx m T_x-n τ$ (pre-revival echoes) and at $t\approx mT_x+nτ$, (post-revival echoes), where $m$ and $n$ are $1,2,...$ . Depending on the form of the perturbations, the 'principal' revival echoes at $t \approx T_x \pm τ$ can be much larger than the echo at $t \approx 2τ$. We develop a perturbative model for these phenomena, and compare its predictions to the numerical solutions of the time-dependent Schrödinger 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 $t=τ$ is investigated. We also study the presence of revivals and revival echoes in higher moments of position, $ $, $p>1$, and the effect of atom-atom interactions on these phenomena.

quant-ph↗

arXiv2012

Statistical Properties of the one dimensional Anderson model relevant for the Nonlinear Schrödinger Equation in a random potential

The statistical properties of overlap sums of groups of four eigenfunctions of the Anderson model for localization as well as combinations of four eigenenergies are computed. Some of the distributions are found to be scaling functions, as expected from the scaling theory for localization. These enable to compute the distributions in regimes that are otherwise beyond the computational resources. These distributions are of great importance for the exploration of the Nonlinear Schrödinger Equation (NLSE) in a random potential since in some explorations the terms we study are considered as noise and the present work describes its statistical properties.

cond-mat.dis-nn↗

arXiv2012

Enhanced transport when Anderson localization is destroyed

We investigate the anomalous transport in optically-induced potentials that are random in both space and time. We find that the time variation destroys Anderson localization, replacing it by transport that is faster than diffusion, which in some cases can be even faster than ballistic. We relate this phenomenon to Chirikov's theory of overlapping resonances, and find radical differences between the anomalously-enhanced transport in one-dimensional and two-dimensional systems.

cond-mat.dis-nn↗

arXiv2012

Is transport in time-dependent random potentials universal ?

The growth of the average kinetic energy of classical particles is studied for potentials that are random both in space and time. Such potentials are relevant for recent experiments in optics and in atom optics. It is found that for small velocities uniform acceleration takes place, and at a later stage fluctuations of the potential are encountered, resulting in a regime of anomalous diffusion. This regime was studied in the framework of the Fokker-Planck approximation. The diffusion coefficient in velocity was expressed in terms of the average power spectral density, which is the Fourier transform of the potential correlation function. This enabled to establish a scaling form for the Fokker-Planck equation and to compute the large and small velocity limits of the diffusion coefficient. A classification of the random potentials into universality classes, characterized by the form of the diffusion coefficient in the limit of large and small velocity, was performed. It was shown that one dimensional systems exhibit a large variety of novel universality classes, contrary to systems in higher dimensions, where only one universality class is possible. The relation to Chirikov resonances, that are central in the theory of Chaos, was demonstrated. The general theory was applied and numerically tested for specific physically relevant examples.

cond-mat.stat-mech↗

arXiv2012

Transport in time-dependent random potentials

The classical dynamics in stationary potentials that are random both in space and time is studied. It can be intuitively understood with the help of Chirikov resonances that are central in the theory of Chaos, and explored quantitatively in the framework of the Fokker-Planck equation. In particular, a simple expression for the diffusion coefficient was obtained in terms of the average power density of the potential. The resulting anomalous diffusion in velocity is classified into universality classes. The general theory was applied and numerically tested for specific examples relevant for optics and atom optics.

cond-mat.stat-mech↗

arXiv2012

Effective noise theory for the Nonlinear Schrödinger Equation with disorder

For the Nonlinear Shrödinger Equation with disorder it was found numerically that in some regime of the parameters Anderson localization is destroyed and subdiffusion takes place for a long time interval. It was argued that the nonlinear term acts as random noise. In the present work the properties of this effective noise are studied numerically. Some assumptions made in earlier work were verified, the dependence of various quantities on the localization length of the linear problem were computed. A scenario for the possible breakdown of the theory for a very long time is outlined.

cond-mat.dis-nn↗

arXiv2011

The Nonlinear Schroedinger Equation with a random potential: Results and Puzzles

The Nonlinear Schroedinger Equation (NLSE) with a random potential is motivated by experiments in optics and in atom optics and is a paradigm for the competition between the randomness and nonlinearity. The analysis of the NLSE with a random (Anderson like) potential has been done at various levels of control: numerical, analytical and rigorous. Yet, this model equation presents us with a highly inconclusive and often contradictory picture. We will describe the main recent results obtained in this field and propose a list of specific problems to focus on, that we hope will enable to resolve these outstanding questions.

math-ph↗

arXiv2011

Eigenvalue repulsion estimates and some applications for the one-dimensional Anderson model

We show that the spacing between eigenvalues of the discrete 1D Hamiltonian with arbitrary potentials which are bounded, and with Dirichlet or Neumann Boundary Conditions is bounded away from zero. We prove an explicit lower bound, given by $Ce^{-bN}$, where $N$ is the lattice size, and $C$ and $b$ are some finite constants. In particular, the spectra of such Hamiltonians have no degenerate eigenvalues. As applications we show that to leading order in the coupling, the solution of a nonlinearly perturbed Anderson model in one-dimension (on the lattice) remains exponentially localized, in probability and average sense for initial conditions given by a unique eigenfunction of the linear problem. We also bound the derivative of the eigenfunctions of the linear Anderson model with respect to a potential change.

cond-mat.dis-nn↗

arXiv2010

Quantum structural phase transition in chains of interacting atoms

A quasi one--dimensional system of trapped, repulsively interacting atoms (e.g., an ion chain) exhibits a structural phase transition from a linear chain to a zigzag structure, tuned by reducing the transverse trap potential or increasing the particle density. Since it is a one dimensional transition, it takes place at zero temperature and therefore quantum fluctuations dominate. In [Fishman, et al., Phys. Rev. B 77, 064111 (2008)] it was shown that the system close to the linear-zigzag instability is described by a $ϕ^4$ model. We propose a mapping of the $ϕ^4$ field theory to the well known Ising chain in a transverse field, which exhibits a quantum critical point. Based on this mapping, we estimate the quantum critical point in terms of the system parameters. This estimate gives the critical value of the transverse trap frequency for which the quantum phase transition occurs, and which has a finite, measurable deviation from the critical point evaluated within the classical theory. A measurement is suggested for atomic systems which can probe the critical trap frequency at sufficiently low temperatures T. We focus in particular on a trapped ion system, and estimate the implied limitations on T and on the interparticle distance. We conclude that the experimental observation of the quantum critical behavior is in principle accessible.

cond-mat.stat-mech↗

arXiv2010

Quantum zigzag transition in ion chains

A string of trapped ions at zero temperature exhibits a structural phase transition to a zigzag structure, tuned by reducing the transverse trap potential or the interparticle distance. The transition is driven by transverse, short wavelength vibrational modes. We argue that this is a quantum phase transition, which can be experimentally realized and probed. Indeed, by means of a mapping to the Ising model in a transverse field, we estimate the quantum critical point in terms of the system parameters, and find a finite, measurable deviation from the critical point predicted by the classical theory. A measurement procedure is suggested which can probe the effects of quantum fluctuations at criticality. These results can be extended to describe the transverse instability of ultracold polar molecules in a one dimensional optical lattice.

cond-mat.stat-mech↗

arXiv2010

Quantum chaos of a mixed, open system of kicked cold atoms

The quantum and classical dynamics of particles kicked by a gaussian attractive potential are studied. Classically, it is an open mixed system (the motion in some parts of the phase space is chaotic, and in some parts it is regular). The fidelity (Lochshmidt echo) is found to exhibit oscillations that can be determined from classical considerations but are sensitive to phase space structures that are smaller than Planck's constant. Families of quasi-energies are determined from classical phase space structures. Substantial differences between the classical and quantum dynamics are found for time dependent scattering. It is argued that the system can be experimentally realized by cold atoms kicked by a gaussian light beam.

quant-ph↗