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Bose-Einstein condensation of a finite number of particles trapped in one or three dimensions Ketterle, W.; van Druten, N.J. General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. Download date: 09 May 2018 Bose-Einstein condensation of a finite number of particles trapped in one or three dimensionsWolfgang Ketterle and N. J. van Druten Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139͑Received 16 January 1996͒Bose-Einstein condensation ͑BEC͒ of an ideal gas is investigated for a finite number of particles. In three dimensions, we find a transition temperature which is lower than in the thermodynamic limit. Lowering the dimension increases the transition temperature and is therefore favorable for BEC. This is in contrast to the standard result obtained in the thermodynamic limit which states that BEC is not possible in, e.g., a onedimensional ͑1D͒ harmonic potential. As a result, 1D atom traps, such as radially tightly confining magnetic traps or optical dipole traps, are promising for studying BEC. ͓S1050-2947͑96͒06807-2͔

General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible.Download date: 10 May 2018 VOLUME 77, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 5 AUGUST 1996Collective Excitations of a (Received 19 June 1996) 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)

General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible.Download date: 24 Mar 2019 VOLUME 77, NUMBER 3 Bose-Einstein condensation of sodium atoms has been observed in a novel "cloverleaf" trap. This trap combines tight confinement with excellent optical access, using only dc electromagnets. Evaporative cooling in this trap produced condensates of 5 3 10 6 atoms, a tenfold improvement over previous results. We measured the condensate fraction and the repulsive mean-field energy, finding agreement with theoretical predictions. [S0031-9007(96) P H Y S I C A L R E V I E W L E T T E R S 15 JULY 1996 Bose-Einstein Condensation in a Tightly Confining dc Magnetic Trap

We investigate the behavior of a weakly interacting nearly one-dimensional (1D) trapped Bose gas at finite temperature. We perform in situ measurements of spatial density profiles and show that they are very well described by a model based on exact solutions obtained using the Yang-Yang thermodynamic formalism, in a regime where other, approximate theoretical approaches fail. We use Bose-gas focusing [Shvarchuck et al., Phys. Rev. Lett. 89, 270404 (2002)] to probe the axial momentum distribution of the gas, and find good agreement with the in situ results.PACS numbers: 03.75. Hh, 05.30.Jp, 05.70.Ce Reducing the dimensionality in a quantum system can have dramatic consequences. For example, the 1D Bose gas with repulsive delta-function interaction exhibits a surprisingly rich variety of physical regimes that is not present in 2D or 3D [1,2]. This 1D Bose gas model is of particular interest because exact solutions for the manybody eigenstates can be obtained using a Bethe ansatz [3]. Furthermore, the finite-temperature equilibrium can be studied using the Yang-Yang thermodynamic formalism [4,5,6], a method also known as the thermodynamic Bethe ansatz. This formalism is the unifying framework for the thermodynamics of a wide range of exactly solvable models. It yields solutions to a number of important interacting many-body quantum systems and as such provides critical benchmarks to condensed-matter physics and field theory [6]. The specific case of the 1D Bose gas as originally solved by Yang and Yang [4] is of particular interest because it is the simplest example of the formalism. The experimental achievement of ultracold atomic Bose gases in the 1D regime [7] has attracted renewed attention to the 1D Bose gas problem [8] and is now providing previously unattainable opportunities to test the Yang-Yang thermodynamics.In this paper, we present the first direct comparison between experiments and theory based on the Yang-Yang exact solutions. The comparison is done in the weakly interacting regime and covers a wide parameter range where conventional models fail to quantitatively describe in situ measured spatial density profiles. Furthermore, we use Bose-gas focusing [9] to probe the equilibrium momentum distribution of the 1D gas, which is difficult to obtain through other means.For a uniform 1D Bose gas, the key parameter is the dimensionless interaction strength γ = mg/ 2 n, where m is the mass of the particles, n is the 1D density, and g is the 1D coupling constant. At low densities or large coupling strength such that γ ≫ 1, the gas is in the strongly interacting or Tonks-Girardeau regime [10]. The opposite limit γ ≪ 1 corresponds to the weakly interacting gas. Here, for temperatures below the degeneracy temperature T d = 2 n 2 /2mk B , one distinguishes two regimes [11]. (i) For sufficiently low temperatures, T ≪ √ γT d , the equilibrium state is a quasi-condensate with suppressed density fluctuations. The system can be treated by the mean-field approach and by the Bogoliubov theory of excitations. The 1D c...

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