PhD thesis in English

5. BEC excitation by modulation of scattering lengthNext, by linearizing the Eqs. (5.3) around the equilibrium widths (5.4), we canobtain information on collective modes related to the BEC shape oscillations. Wewill consider experimentally relevant case of an axially symmetric trap, such thatλ x = λ y = 1, u x0 = u y0 ≡ u ρ0 . Due to the axial symmetry of the consideredsystem, the projection of the angular momentum along the z-axis, L z , is a goodquantum number and we use it for the classification of the modes. By solvingthe corresponding eigenproblem, we find three different modes depicted in Fig. 5.2.According to their symmetry properties, we designate the modes as the |L z | = 2quadrupole mode, the L z = 0 quadrupole mode, and the breathing mode (whichalso corresponds to L z = 0). Eigenvalues of the linearized system of equations yieldthe frequencies of the collective modes. Frequencies of the L z = 0 quadrupole modeω Q0 , and the breathing mode ω B0 are given by:ω B0,Q0 = √ 2[ (1 + λ 2 z − P4u 2 ρ0 u3 z0)√ (± 1 − λ 2 z +P4u 2 ρ0 u3 z0while the frequency of the |L z | = 2 quadrupole mode is given by:ω |Lz|=2Q0=) 2 ( )]1/2 2 P+ 8,4u 3 ρ0 u2 z0(5.5)√P4 − 2 . (5.6)u 4 ρ0u z0As shown in Fig. 5.2, the |L z | = 2 quadrupole mode is characterized by out-ofphaseoscillations in x and y directions, the |L z | = 0 quadrupole mode exhibitsout-of-phase radial and axial oscillations, while in-phase radial and axial oscillationscorrespond to the breathing mode.Figure 5.2: A schematic illustration of the condensate eigenmodes: |L z | = 2quadrupole mode (left), |L z | = 0 quadrupole mode (middle) and |L z | = 0 breathingmode (right).110