PhD thesis in English

separate treatment.5. BEC excitation by modulation of scattering lengthIn Ref. [116] it was predicted that a harmonic modulation of the scattering lengthleads to the creation of Faraday patterns in BEC, i.e. density waves. Up to now,Faraday patterns have been experimentally induced by harmonic modulation of thetransverse confinement strength [111], which is studied analytically and numericallyin Ref. [129]. Here we focus only on the nonlinear properties of low-lying collectivemodes and do not consider possible excitations of Faraday patterns.In order to obtain analytical insight into the condensate dynamics induced bythe harmonic modulation of the s-wave scattering length described by Eq. (5.7), weuse the Gaussian variational approximation approximation. We consider an axiallysymmetric BEC, (λ x = λ y = 1, u x = u y ≡ u ρ ), excited by modulation of theinteraction strength, which preserves the axial symmetry of the condensate duringits time evolution. For this reason, we use a simplified axially-symmetric form ofEqs. (5.3):ü ρ (t) + u ρ (t) − 1u ρ (t) 3 −P(t)u ρ (t) 3 u z (t)= 0 , (5.11)ü z (t) + λ 2 zu z (t) − 1u z (t) − P(t)= 0 , (5.12)3 u ρ (t) 2 u z (t) 2which we will refer to as a Gaussian approximation.To estimate the accuracy of the Gaussian approximation for describing the dynamicsinduced by the harmonic modulation of the interaction strength, we compareits solution with an exact numerical solution of the GP equation. In Fig. 5.5, weplot the resulting time-dependent axial and radial condensate widths ρ rms (t) andz rms (t), calculated as root mean square valuesρ rms (t) =z rms (t) =√√2π2π∫ ∞−∞∫ ∞−∞dzdz∫ ∞0∫ ∞0ρ dρ |ψ(ρ, z, t)| 2 ρ 2 , (5.13)ρ dρ |ψ(ρ, z, t)| 2 z 2 , (5.14)of the solution of the GP equation, as well as numerical solutions of Eqs. (5.11)and (5.12). We assume that initially the condensate is in the ground state. In thevariational description, this translates into initial conditions u ρ (0) = u ρ0 , ˙u ρ (0) = 0,u z (0) = u z0 , ˙u z (0) = 0, where u ρ0 and u z0 are the time-independent solutions115