PhD thesis in English

4. Mean-field description of an interacting BEC4.1 Gross-Pitaevskii equationIn the zero-temperature limit, we can neglect the thermal cloud, and setlim n th(⃗r) = 0 , (4.12)β→∞which stems directly from Eq. (4.10). In this case, Eq. (4.11) becomes irelevant,while from Eq. (4.9) we find the time-independent Gross-Pitaevskii (GP) equation[13, 14, 34] for the order parameter:[− 22M △ + V (⃗r) + g|ψ(⃗r)|2 ]ψ(⃗r) = µψ(⃗r) . (4.13)Effectively, in the mean-field approximation at zero temperature, we assume thatall atoms occupy the same state ψ(⃗r), which we denote as the condensate wavefunction.Note that already by neglecting anomalous averages we have disregardeda depletion of the condensate at zero temperature that arises due to interactions.However, it turns out that this is a reasonable approximation in the wide range ofexperimental parameters for a weakly interacting gas. On the left-hand side of theGP equation (4.13) we have a kinetic energy term, an external trap potential V (⃗r),and a nonlinear term originating from the mean-field interaction between the atoms.The GP equation belongs to the class of nonlinear Schrödinger equations, which areextensively studied also in the field of nonlinear optics [85, 86].Now, let us analyze solutions of the GP equation. To begin with, we note thatthe noninteracting limit is straightforwardly reproduced: for g → 0, the condensatewavefunction is the ground state of the external potential V (⃗r), and the value of thechemical potential is equal to the ground-state energy. In the limit of strong repulsiveinteractions (for a large number of atoms, for example), the term correspondingto the kinetic energy can be safely neglected. In that case we find an algebraicstationary solution|ψ(⃗r)| 2 = 1 (µ − V (⃗r)) θ(µ − V (⃗r)), (4.14)gwhich is the well-known Thomas-Fermi (TF) solution [19]. The value of the chemicalpotential µ is determined as usual, by fixing the total number of atoms in the systemto N. In particular, for the harmonic trap, the solution for the noninteracting case isa Gaussian, while the TF solution yields a parabolic profile. It is easy to understand88