Measuring the heat capacity in a Bose-Einstein condensation using global variables (original) (raw)
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Nuclear Physics A, 2007
We have used the definition of global thermodynamic variables such as generalized pressure and generalized volume for atoms trapped in a non-uniform potential to evaluate their behavior during the crossing of the critical temperature during a Bose-Einstein condensation. Comparing the experimental results with a theory based on a Hartree-Fock method, we have shown the regime of validity for the approximation. * Financial support from DGAPA IN-117406-2 (Mexico). † Financial support from Fapesp, CNPq and CAPES (Brasil).
Thermodynamics of a Trapped Bose-Condensed Gas
Journal of Low Temperature Physics, 1997
with repulsive forces and confined in a harmonic anisotropic trap. We develop the formalism of mean field theory for non uniform systems at finite temperature, based on the generalization of Bogoliubov theory for uniform gases. By employing the WKB semiclassical approximation for the excited states we derive systematic results for the temperature dependence of various thermodynamic quantities: condensate fraction, density profiles, thermal energy, specific heat and moment of inertia. Our analysis points out important differences with respect to the thermodynamic behaviour of uniform Bose gases. This is mainly the consequence of a major role played by single particle states at the boundary of the condensate. We find that the thermal depletion of the condensate is strongly enhanced by the presence of repulsive interactions and that the critical temperature is decreased with respect to the predictions of the non-interacting model. Our work points out an important scaling behaviour exhibited by the system in large N limit. Scaling permits to express all the relevant thermodynamic quantities in terms of only two parameters: the
Trapped Bose-einstein Condensates at Finite Temperature: a Two-gas Model
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A simple picture describes the results of recent treatments of partiallycondensed, dilute, trapped Bose gases at temperature T > 0. The condensate wavefunction is nearly identical to that of a T = 0 condensate with the same number of condensate atoms, N 0. The cloud of non-condensed atoms is described by the statistical mechanics of an ideal Bose gas in the combined potentials of the magnetic trap and the cloud-condensate interaction. We provide a physical motivation for this result, show how it emerges in the Hartree-Fock-Bogoliubov-Popov approximation, and explore some of its implications for future experiments.
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This paper presents the results of our recent experiments on the finite-temperature Bose-Einstein condensate of 87 Rb atoms in a magnetic trap, and is devoted to the study of the hydrodynamic properties and dynamics of an ultra-cold atomic gas near the critical temperature. Measurements of the aspect ratio of an expanding atomic cloud allow for verification of the condensate models and study of the interaction between condensed and non-condensed fractions of a finite-temperature sample.
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Phase transitions, as the condensation of a gas to a liquid, are often revealed by a discontinuous behaviour of thermodynamic quantities. For liquid helium, for example, a divergence of the specific heat signals the transition from the normal fluid to the superfluid state. Apart from liquid helium, determining the specific heat of a Bose gas has proven to be a challenging task, for example, for ultracold atomic Bose gases. Here we examine the thermodynamic behaviour of a trapped two-dimensional photon gas, a system that allows us to spectroscopically determine the specific heat and the entropy of a nearly ideal Bose gas from the classical high temperature to the Bose-condensed quantum regime. The critical behaviour at the phase transition is clearly revealed by a cusp singularity of the specific heat. Regarded as a test of quantum statistical mechanics, our results demonstrate a quantitative agreement with its predictions at the microscopic level.
Behavior of heat capacity of an attractive Bose-Einstein condensate approaching collapse
We report the calculation of heat capacity of an attractive Bose-Einstein condensate, with the number N of bosons increasing and eventually approaching the critical number Ncr for collapse, using the correlated potential harmonics (CPH) method. Boson pairs interact via the realistic van der Waals potential. It is found that the transition temperature Tc initially increases slowly, then rapidly as N becomes closer to Ncr . The peak value of heat capacity for a fixed N increases slowly with N , for N far away from Ncr . But after reaching a maximum, it starts decreasing when N approaches Ncr . The effective potential calculated by the CPH method provides insight into this strange behavior.
Momentum distribution of an interacting Bose-Einstein condensed gas at finite temperature
Physical Review A, 2000
We use a semiclassical two-fluid model to study the momentum distribution of a Bose-condensed gas with repulsive interactions inside a harmonic trap at finite temperature, with specific focus on atomic hydrogen. We give particular attention to the average kinetic energy, which is almost entirely associated with the thermal cloud. A non-linear dependence of the kinetic energy on temperature is displayed, affording a precise way to assess the temperature of the gas. We also show that the kinetic energy increases with the strength of the interactions, reflecting an enhanced rate of depletion of the condensate with increasing temperature.
Theory of Bose-Einstein condensation in trapped gases
Reviews of Modern Physics, 1999
The phenomenon of Bose-Einstein condensation of dilute gases in traps is reviewed from a theoretical perspective. Mean-field theory provides a framework to understand the main features of the condensation and the role of interactions between particles. Various properties of these systems are discussed, including the density profiles and the energy of the ground state configurations, the collective oscillations and the dynamics of the expansion, the condensate fraction and the thermodynamic functions. The thermodynamic limit exhibits a scaling behavior in the relevant length and energy scales. Despite the dilute nature of the gases, interactions profoundly modify the static as well as the dynamic properties of the system; the predictions of mean-field theory are in excellent agreement with available experimental results. Effects of superfluidity including the existence of quantized vortices and the reduction of the moment of inertia are discussed, as well as the consequences of coherence such as the Josephson effect and interference phenomena. The review also assesses the accuracy and limitations of the mean-field approach.
Condensate fraction and critical temperature of a trapped interacting Bose gas
Physical Review A, 1996
By using a mean field approach, based on the Popov approximation, we calculate the temperature dependence of the condensate fraction of an interacting Bose gas confined in an anisotropic harmonic trap. For systems interacting with repulsive forces we find a significant decrease of the condensate fraction and of the critical temperature with respect to the predictions of the non-interacting model. These effects go in the opposite direction compared to the case of a homogeneous gas. An analytic result for the shift of the critical temperature holding to first order in the scattering length is also derived.
Thermodynamics of Bose–Einstein gas trapped in -dimensional quartic potentials
Physica A: Statistical Mechanics and its Applications, 2010
Quartic potentials play an important role in a broad range of fields in the Bose-Einstein Condensation (BEC) literature, from optical lattices to vortices and even to the quantum phase transitions. This also makes the study of thermodynamic properties of systems confined by these potentials interesting. For this purpose, the BEC of an ideal Bose system with a finite number of particles is considered in three and lower dimensions. A comprehensive thermodynamic analysis including the condensation temperature, chemical potential, condensate fraction, heat capacity, total energy, and entropy has been carried out with a special emphasis on low-dimensional case for which the standard semi-classical method is not applicable. As a result of our calculations based on a quantum mechanical treatment we have shown that BEC is possible in a one-dimensional quartic potential, in contrast with the predictions of the standard method. Moreover it is also possible to corroborate our results by using a modified semi-classical approximation that enables us to estimate the condensation temperatures for one-dimensional traps for all powers of the confining potential. We have found higher condensation temperatures than for the harmonic traps for all the three spatial configurations of the quartic trap.