PhD thesis in English

5. BEC excitation by modulation of scattering lengthmodes obtained previously in the linear regime. As explained in Chapter 4, thetime-dependent GP equation can be obtained by extremizing the functional (4.16)with respect to ψ(⃗r, t). In the core of the variational description is the idea to plugin an Ansatz for the condensate wave function into Eq. (4.16) and to derive the correspondingEuler-Lagrange equations of the system with respect to the parameterspresent in the Ansatz. In this way, instead of the partial differential GP equation,we reduce the description of a system to ordinary differential equations, which is farsimpler. Naturally, this presents an approximation, and careful examination of itsvalidity and associated errors is necessary.To this end, we closely follow derivations presented originally in Refs. [104, 105].We assume that the condensate wave function has the same Gaussian form in theinteracting case as in the noninteracting one, just with renormalized parameters.Thus, we use a time-dependent variational method based on a Gaussian Ansatz,which for the anisotropic harmonic trap (1.19) readsψ G (x, y, z, t) = N(t)∏σ=x,y,z[exp − 1 ](σ − σ 0 (t)) 2+ i σϕ2 u σ (t) 2 σ (t) + iσ 2 φ σ (t) , (5.1)where N(t) = π −3 4u x (t) −1 2u y (t) −1 2u z (t) −1 2 is a time-dependent normalization, whileu σ (t), φ σ (t), σ 0 and ϕ σ are variational parameters. For convenience, throughoutthis Chapter, we normalize the wave function to unity and for consistency we includethe total number of atoms into the corresponding interaction strength. Thus,we modify the notation introduced earlier by performing the following transformation:ψ(⃗r, t) → ψ(⃗r, t)/ √ N, g → g × N. The introduced variational parametershave straightforward interpretation: u σ (t) parameters correspond to the condensatewidths in different directions and are roughly proportional to the root-mean-squarewidths of the exact condensate wave function ψ(x, y, z, t); φ σ (t) and ϕ σ (t) parametersrepresent the corresponding phases of the wave function and are essential forthe proper description of dynamical features; a possible center-of-mass motion iscaptured by the parameters σ 0 .Following Ref. [104], we insert Ansatz (5.1) into the Lagrangian (4.16) yieldingthe GP equation, and extremize it with respect to variational parameters. All thedetails of the derivation are given in Appendix B and here we give only a briefexplanation. By extremizing the functional, we first obtain a coupled system ofdifferential equations of the first order for all variational parameters. The equations108