1,721,100 research outputs found
Accelerated Bose-Einstein condensates in a double-well potential
No full textDevices based on ultracold atoms moving in an accelerating optical lattice or double-well potential are a promising tool for precise measurements of fundamental physical constants as well as for the construction of sensors. Here, we carefully analyze the model of a couple of BECs separated by a barrier in an accelerated field and we show how the observable quantities, mainly the period of the beating motion or of the phase-shift, are related to the physical parameters of the model as well as to the energy of the initial state
Double-barrier resonances and time decay of the survival probability: A toy model
No full textIn this talk we consider the time evolution of a one-dimensional quantum system with a double barrier given by a couple of repulsive Dirac’s deltas. In such a pedagogical model we give, by means of the theory of quantum resonances, the asymptotic behavior of (Formula Found) for large times, where H is the double-barrier Hamiltonian operator and where ψ and φ are two test functions. In particular, when ψ is close to a resonant state then explicit expression of the dominant terms of the survival probability defined as (Formula Found) is given
Nonlinear Schrödinger equations with multiple-well potential
No full textWe consider the stationary solutions for a class of Schrödinger equations with a N-well potential and
a nonlinear perturbation. By means of semiclassical techniques we prove that the dominant term of
the ground state solutions is described by a N-dimensional Hamiltonian system, where the coupling
term among the coordinates is a tridiagonal Toeplitz matrix. In particular, in the limit of large focusing
nonlinearity we prove that the ground state stationary solutions consist of N wavefunctions localized on
a single well. Furthermore, we consider in detail the case of N = 4 wells, where we show the occurrence
of spontaneous symmetry-breaking bifurcation effect
Stationary solutions to the multi-dimensional Gross–Pitaevskii equation with double-well potential
No full textIn this paper we consider a nonlinear Schr ̈odinger equation with a cubic
nonlinearity and a multi-dimensional double well potential. In the semiclassical
limit the problem of the existence of stationary solutions simply reduces to the
analysis of a finite dimensional Hamiltonian system which exhibits different
behaviour depending on the dimension. In particular, in dimension 1 the
symmetric stationary solution shows a standard pitchfork bifurcation effect,
while in dimensions 2 and 3 new asymmetrical solutions associated with saddle
points occur. These last solutions are localized on a single well and this fact is
related to the phase transition effect observed in Bose–Einstein condensates in
periodical lattices
Nonlinear Schrödinger equations with a multiple-well potential and a Stark-type perturbation
Get PDFA Bose-Einstein condensate (BEC) confined in a one-dimensional lattice under the effect of an external homogeneous field is described by the Gross-Pitaevskii equation. Here we prove that such an equation can be reduced, in the semiclassical limit and in the case of a lattice with a finite number of wells, to a finite-dimensional discrete nonlinear Schrödinger equation. Then, by means of numerical experiments we show that the BEC's center of mass exhibits an oscillating behavior with modulated amplitude; in particular, we show that the oscillating period actually depends on the shape of the initial wavefunction of the condensate as well as on the strength of the nonlinear term. This fact opens a question concerning the validity of a method proposed for the determination of the gravitational constant by means of the measurement of the oscillating period
Spectral splitting method for nonlinear Schrodinger equations with singular potential
No full textWe consider the time-dependent one-dimensional nonlinear Schro ̈dinger equation with pointwise singular potential. Bymeans of spectral splitting methods we prove that the evolution operator is approximated by the Lie evolution operator,where the kernel of the Lie evolution operator is explicitly written. This result yields a numerical procedure which is muchless computationally expensive than multi-grid methods previously used. Furthermore, we apply the Lie approximation inorder to make some numerical experiments concerning the splitting of a soliton, interaction among solitons and blow-upphenomenon
Universal Critical Power for Nonlinear Schrödinger Equations with a Symmetric Double Well Potential
No full textHere we consider stationary states for nonlinear Schrödinger equations in any spatial dimension n with symmetric double well potentials. These states may bifurcate as the strength of the nonlinear term increases and we observe two different pictures depending on the value of the nonlinearity power: a supercritical pitchfork bifurcation, and a subcritical pitchfork bifurcation with two asymmetric branches occurring as the result of saddle-node bifurcations. We show that in the semiclassical limit, or for a large barrier between the two wells, the first kind of bifurcation always occurs when the nonlinearity power is less than a critical value; in contrast, when the nonlinearity power is larger than such a critical value then we always observe the second scenario. The remarkable fact is that such a critical value is a universal constant in the sense that it does not depend on the shape of the double well potential and on the dimension n
Second School and Workshop on "Mathematical Methods in Quantum Mechanics"
No full textAim and topicsThe aim of the meeting is to present the state of the art in some challenging open problems in Quantum Mechanics from the point of view of Mathematical Physics. It is mainly addressed to young people interested in working on the subject. Among the topics covered: Derivation of macroscopic equations from microscopic quantum dynamics, coupled dynamics of particles and radiation fields, quantum information and entanglement, classical behaviour in quantum systems, scattering and spectral analysis for Schrödinger operators, quantum graphs.Three short courses will be given in a series of lectures scheduled in the morning of each day. Some invited talks will be given in the afternoon followed by short contributed talks given by participants
Third School and Workshop on "Mathematical Methods in Quantum Mechanics"
No full textThe aim of the meeting is to present the state of the art in some challenging open problems in Quantum Mechanics from the point of view of Mathematical Physics. It is mainly addressed to young people interested in working on the subject.Among the topics covered: quantum systems with magnetic fields, quantum transport theory, quantum dechoerence and entanglement, classical behaviour in quantum systems, scattering and spectral analysis for Schroedinger operators, quantum chaos, adiabatic and semiclassical methods. non linear Schroedinger equations.Three courses will be given in a series of lectures scheduled in the morning of each day. Some invited talks will be given in the afternoon followed by short contributed talks given by participants
Nonlinear double-well Schrodinger equations in the semiclassical limit
No full textWe consider time-dependent Schrodinger equations with a double well potential and an external nonlinear, both local and non-local, perturbation. In the semiclassical limit, the finite dimensional eigenspace associated to the lowest eigenvalues of the linear operator is almost invariant for times of the order of the beating period and the dominant term of the wavefunction is given by means of the solutions of a finite dimensional dynamical system. In the case of local nonlinear perturbation, we assume the spatial dimension d=1 or d=2
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