PhD thesis in English

5. BEC excitation by modulation of scattering lengthfor the phases φ σ and ϕ σ can be solved explicitly in terms of the widths u σ andthe center of mass coordinates σ 0 . In this way we obtain two sets of ordinarysecond-order differential equations that govern the condensate dynamics. A centerof-massmotion is decoupled from the shape oscillations and is determined by simpleharmonic oscillator equations, which are independent of interatomic interactions:¨σ 0 (t) + λ 2 σσ 0 (t) = 0, σ ∈ {x, y, z} . (5.2)On the other hand, the widths of the condensate exhibit non-trivial dynamics givenby a set of coupled nonlinear differential equations:ü σ (t) + λ 2 σu σ (t) − 1u σ (t) − P= 0, σ ∈ {x, y, z} . (5.3)3 u σ (t)u x (t)u y (t)u z (t)In the previous equations and from now on, we use dimensionless notation: wechoose a convenient frequency scale ω (for example, the external trap frequency inone of the spatial directions) and express all lengths in the units of the characteristicharmonic oscillator length l = √ /Mω, time in units of ω −1 and external frequenciesin units of ω: λ σ = ω σ /ω, σ ∈ {x, y, z}. The dimensionless interaction parameter Pis given by P = g/((2π) 3/2 ωl 3 ) = √ 2/πNa/l.In this approach, the initial partial differential equation (4.15) is approximatedwith the two sets of ordinary differential equations, given by Eqs. (5.2) and (5.3),which allow analytical considerations. The first properties of the condensate thatcan be calculated are the equilibrium widths u x0 , u y0 and u z0 . They are found bysolving an algebraic system of equations:λ 2 σu σ0 − 1 P− = 0, σ ∈ {x, y, z} . (5.4)u σ0 u σ0 u x0 u y0 u z0The equilibrium widths represent stationary solutions of Eqs. (5.3).Now we turn to the calculation of the frequencies of low-lying modes. To beginwith, from Eqs.(5.2) we read off frequencies that correspond to the center-of-massmotion. These are dipole modes and their frequencies are equal to the external trapfrequencies (for the case of a harmonic trap). Most often, this type of excitationsis created by shifting the trap in space. Actually, the well established experimentalprocedure for the precise determination of the trap parameters is based on themeasurement of the dipole mode frequencies [21].109