379 research outputs found
Accelerated Bose-Einstein condensates in a double-well potential
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
We consider the Gross-Petaevskii equation in 1 space dimension with a
-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
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
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
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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