379 research outputs found

    Accelerated Bose-Einstein condensates in a double-well potential

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    Devices 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.Comment: 5 figure

    Exponential times in the one-dimensional Gross--Petaevskii equation with multiple well potential

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    We consider the Gross-Petaevskii equation in 1 space dimension with a nn-well trapping potential. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest n eigenvalues of the linear operator is slightly deformed by the nonlinear term into an almost invariant manifold M. Precisely, one has that solutions starting on M, or close to it, will remain close to M for times exponentially long with the inverse of the size of the nonlinearity. As heuristically expected the effective equation on M is a perturbation of a discrete nonlinear Schroedinger equation. We deduce that when the size of the nonlinearity is large enough then tunneling among the wells essentially disappears: that is for almost all solutions starting close to M their restriction to each of the wells has norm approximatively constant over the considered time scale. In the particular case of a double well potential we give a more precise result showing persistence or destruction of the beating motions over exponentially long times. The proof is based on canonical perturbation theory; surprisingly enough, due to the Gauge invariance of the system, no non-resonance condition is required

    Stability of spectral eigenspaces in nonlinear Schrodinger equations

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    We consider the time-dependent non linear Schrodinger equations with a double well potential in dimensions d =1 and d=2. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest two eigenvalues of the linear operator is almost invariant for any time

    Perturbation theory for nonlinear Schrödinger equations

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    Treating the nonlinear term of the Gross–Pitaevskii nonlinear Schrödinger equation as a perturbation of an isolated discrete eigenvalue of the linear problem one obtains a Rayleigh–Schrödinger power series. This power series is proved to be convergent when the parameter representing the intensity of the nonlinear term is less in absolute value than a threshold value, and it gives a stationary solution to the nonlinear Schrödinger equation

    Derivation of the Tight-Binding Approximation for Time-Dependent Nonlinear Schr\uf6dinger Equations

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    In this paper, we consider the nonlinear one-dimensional timedependent Schr\ua8odinger equation with a periodic potential and a bounded perturbation. In the limit of large periodic potential, the time behavior of the wavefunction can be approximated, with a precise estimate of the remainder term, by means of the solution to the discrete nonlinear Schroedinger equation of the tight-binding model
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