Collective Excitations of a Bose-Einstein Condensate in a Magnetic Trap.

Collective excitations of a dilute Bose condensate have been observed. These excitations are analogous to phonons in superfluid helium. Bose condensates were created by evaporatively cooling magnetically trapped sodium atoms. Excitations were induced by a modulation of the trapping potential, and detected as shape oscillations in the freely expanding condensates. The frequencies of the lowest modes agreed well with theoretical predictions based on mean-field theory. Before the onset of BoseEinstein condensation, we observed sound waves in a dense ultracold gas. [S0031-9007(96)00900-3] In 1941 Landau introduced the concept of elementary excitations to explain the properties of superfluid helium [1]. This phenomenological approach, based on quantum hydrodynamics, gave a quantitative description of the thermodynamic properties and transport processes in liquid helium. Landau rejected any relation to Bose-Einstein condensation (BEC). A microscopic derivation of the elementary excitation spectrum for a weakly interacting Bose gas was given by Bogoliubov in 1947 [2] and for He II by Feynman in 1955 [3], emphasizing the role of Bose statistics [3] and reconciling Landau’s approach with London’s explanation of superfluidity as being due to BEC [2,4]. The elementary excitations determine the spectrum of density fluctuations in a Bose liquid, and have been directly observed in He II by neutron scattering [5]. The low-frequency excitations are phonons, long-wavelength collective modes of the superfluid. So far, a satisfactory microscopic theory for an interacting bosonic system exists only for the dilute quantum gas. The recent realization of BEC in dilute atomic vapors [6 ‐ 8] has opened the door to test this theory experimentally. In this paper we report on the observation of shape oscillations of a trapped Bose condensate, modes analogous to phonons in homogeneous systems [9]. The experimental setup for creating Bose condensates was the same as in our previous work [10]. Briefly, sodium atoms were optically cooled and trapped, and transferred into a magnetic trap where they were further cooled by rf-induced evaporation [11,12]. Every 30 s, condensates containing 5 3 10 6 sodium atoms in the F › 1, mF › 21 ground state were produced. Evaporative cooling was extended well below the transition temperature to obtain a condensate without a discernible normal component. The condensate was confined in a cloverleaf magnetic trap which had cylindrical symmetry with trapping frequencies of 19 Hz axially and 250 Hz radially (see below). The trapping potential is determined by the axial curvature of the magnetic field B 00 › 125 Gc m 2 2 , the radial gradient B 0 › 150 Gc m 2 1 , and the bias field B0 › 1.2 G. The condensate was excited by a time-dependent modulation of the trapping potential. First, we used a sudden step in the gradient B 0 to identify several collective modes of the condensate and to find their approximate frequencies. B 0 was decreased by 15% for a duration of 5 ms with a transition time of about 1 ms, and then returned to its original value. A variable time delay was introduced between the excitation and the observation of the cloud. In this way, we strobed the free time evolution of the system after the excitation. The cloud was observed by absorption imaging after a sudden switch off of the magnetic trap and 40 ms of ballistic expansion. No trap loss was observed during the interval over which the delay was varied. The images were similar to the series shown in Fig. 1. Four modes were identified from the measured center-of-mass positions and the widths of the condensate. The radial and axial center-of-mass oscillations (dipole modes) were excited because a change in B 0 displaced the center of the trap slightly due to asymmetries in the field-producing coils. A fast shape oscillation predominantly showed up as a sinusoidal modulation of the radial width while a slow sinusoidal shape oscillation was observed in the axial width. When a strong parametric drive (see below) was used to excite the slow shape oscillation, a weak oscillation of the radial width was also detected. Note that the widths were observed after ballistic expansion and reflect a convolution of the initial spatial and velocity