We present analytical results for the macroscopic wave function of a Bose-Einstein condensate in a time dependent harmonic potential. The evolution of the spatial density is a dilatation, characterized by three scaling factors which allow a classical interpretation of the dynamics. This approach is an efficient tool for the analysis of recent experimental results on the expansion and collective excitation of a condensate. [S0031-9007 (96)01919-9] PACS numbers: 03.75.Fi, 05.30.JpRecently the combination of laser cooling and evaporative cooling led to the observation of Bose-Einstein condensation in dilute atomic vapors [1][2][3]. The favored observation technique has been a time of flight measurement: the trapping potential is rapidly switched off and the spatial distribution of the expanding cloud is monitored. In more recent experiments the condensates were collectively excited by a time modulation of the trapping potential [4,5]. In these experiments the state of the condensate is strongly influenced by atomic interactions, which must therefore be included in a theoretical treatment. Work up to now consisted in the numerical solution of the time dependent nonlinear Schrödinger equation for the macroscopic wave function of the condensate [6]. We present here analytical results which allow a more lucid description of the condensate dynamics and an immediate comparison with experiment. To this end we introduce a quantum scaling transform [7] which is inspired by a model of a classical gas. Applying our results to time of flight measurements of expanding condensates [3] we obtain the scattering length of sodium. For condensates collectively excited by a time modulation of the trapping potential we present an ab initio calculation of the observed signal.For dilute gases at low temperatures the atomic interactions can be modeled by a pseudopotential gd͑ r͒, where g . 0 is related to the s-wave scattering length a by g 4ph 2 a͞m [8]. We describe the trap by an anisotropic time dependent harmonic potentialWe restrict the discussion to the case of zero temperature, which is a realistic assumption for the experiments in [1,2]. The state of the condensate for a static trap can thus be described using a Hartree-Fock ansatz:jC͘ jF͘ ≠ · · · ≠ jF͘ .The minimization of mean energy gives the time independent Gross-Pitaevskii equation for jF͘:with N 2 1 Ӎ N.In the regime where the atomic interactions are dominant [NgjF͑ 0͒j 2 Ӎ m ¿hv j for j 1, 2, 3] we can use the Thomas-Fermi approximation to solve (3) [9]; that is, we can neglect the kinetic energy term as compared to the interaction energy term. The result iswhen m $ U͑ r, 0͒, and F͑ r͒ 0 otherwise. The chemical potential m is determined by the normalization of jF͘:wherev ͓v 1 ͑0͒v 2 ͑0͒v 3 ͑0͔͒ 1͞3 .

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We investigate minimal energy solutions with vortices for an interacting Bose-Einstein condensate in a rotating trap. The atoms are strongly confined along the axis of rotation z, leading to an effective 2D situation in the x−y plane. We first use a simple numerical algorithm converging to local minima of energy. Inspired by the numerical results we present a variational Ansatz in the regime where the interaction energy per particle is stronger than the quantum of vibration in the harmonic trap in the x − y plane, the so-called Thomas-Fermi regime. This Ansatz allows an easy calculation of the energy of the vortices as function of the rotation frequency of the trap; it gives a physical understanding of the stabilisation of vortices by rotation of the trap and of the spatial arrangement of vortex cores. We also present analytical results concerning the possibility of detecting vortices by a time-of-flight measurement or by interference effects. In the final section we give numerical results for a 3D configuration.