Bifurcation trees of Stark-Wannier ladders for accelerated Bose-Einstein condensates in an optical lattice

Abstract

In this paper we show that in the semiclassical regime of periodic potential large enough, the Stark-Wannier ladders become a dense energy spectrum because of a cascade of bifurcations while increasing the ratio between the effective nonlinearity strength and the tilt of the external field; this fact is associated to a transition from regular to quantum chaotic dynamics. The sequence of bifurcation points is explicitly given.

Nonlinear Stark--Wannier Equation

“…Let S be a solution-set to (26) with associated energy µ S and normalized stationary solution d S given by (28). We assume that S ⊂ [−N, N ].…”

“…To this end we must introduce some technical assumptions on W , that is W must be a locally linear bounded function with compact support; in fact in the case of a true Stark potential where W (x) = x some basic estimates useful in our analysis don't work because W is not a bounded operator. Some results, like the occurrence of a cascade of bifurcations for the discrete nonlinear Schrödinger equation in the anticontinuous limit has been already announced in a physics-oriented paper [28] without mathematical details. We should also mention a recent paper [14] where bifurcations are observed in rotating Bose-Einstein condensates.…”

Derivation of the Tight-Binding Approximation for Time-Dependent Nonlinear Schrödinger Equations

“…Indeed, the perturbative terms with strength F and η dominate the coupling term with strength β between the adjacent wells. In fact, this model has some interesting features; for instance, when W represents a Stark-type perturbation then the analysis of the stationary solutions exhibits the existence of a cascade of bifurcations [22,23]. On other hand, due to the fact that the perturbation is large, when compared with the coupling term, the validity of the tight-binding approximation is justified only for time intervals rather small.…”

“…(27)Then, we get the integral inequalityc − g ℓ 2 ≤ α(τ ) + τ ) ≤ C max[h −1 β, λ]By the Gronwall's Lemma we finally get the estimatec − g ℓ 2 ≤ α(τ )e τ 0 δ(q)dq = α(τ )e C max[h −1 β,λ]τTherefore, we have proved that Lemma 10. Let a, b and c dafined by(25)(26)(27), then Equazioni Differenziali della Fisica Matematicac − g ℓ 2 ≤ C aτ h + Cbλ + c C 2 λ 2 h e Cλτ − 1 + cτ Cλh e Cλτ e C max[h −1 β,λ]τ (29)for some positive constant C independent of h.In conclusion Let g ∈ C(R, ℓ 2 (Z)) be the solution to the discrete nonlinear Schrödinger equation(22); let ψ(τ, x) ∈ C(R, H 1 (R)) be the solution to the nonlinear Schrödinger equation (3) with initial condition ψ 0 (x) = g n (0)u n (x); let a, b and c defined by Lemma 10; let λ be defined by Lemma 7. Then, for some positive constant C independent of h it follows thatψ(τ, •) − n∈Z g n (τ )e iΛ1τ /h u n (•) L (30) ≤ Cλ + τ Ch −1 λ max [β, hλ] e Cλτ + +C aτ h + Cbλ + c C 2 λ 2 h e Cλτ − 1 + cτ Cλh e Cλτ e C max[h −1 β,λ]τ , ∀τ ∈ R + .(31)Proof.…”