Using in situ measurements on a quasi-two-dimensional, harmonically trapped (87)Rb gas, we infer various equations of state for the equivalent homogeneous fluid. From the dependence of the total atom number and the central density of our clouds with chemical potential and temperature, we obtain the equations of state for the pressure and the phase-space density. Then, using the approximate scale invariance of this 2D system, we determine the entropy per particle and find very low values (below 0.1k(B)) in the strongly degenerate regime. This shows that this gas can constitute an efficient coolant for other quantum fluids. We also explain how to disentangle the various contributions (kinetic, potential, interaction) to the energy of the trapped gas using a time-of-flight method, from which we infer the reduction of density fluctuations in a nonfully coherent cloud.
Using in situ measurements on a quasi-two-dimensional, harmonically trapped (87)Rb gas, we infer various equations of state for the equivalent homogeneous fluid. From the dependence of the total atom number and the central density of our clouds with chemical potential and temperature, we obtain the equations of state for the pressure and the phase-space density. Then, using the approximate scale invariance of this 2D system, we determine the entropy per particle and find very low values (below 0.1k(B)) in the strongly degenerate regime. This shows that this gas can constitute an efficient coolant for other quantum fluids. We also explain how to disentangle the various contributions (kinetic, potential, interaction) to the energy of the trapped gas using a time-of-flight method, from which we infer the reduction of density fluctuations in a nonfully coherent cloud.
Thermodynamics of non-uniform systems has always been a challenging topic in the physical sciences.In the field of quantum gases, that subject is mainly studied with ultracold samples trapped in a harmonic potential, the first kind of confinement ever made to atomic fluids. 1 Nascimbène et al. 2 have already described a method to measure locally the equation of state and heat capacity of a harmonically trapped quantum gas.Nonetheless, local measurements cannot provide the volume-dependent susceptibilities of an inhomogeneous system, namely the thermal expansion and compressibility, meaning that the thermodynamic description of harmonically trapped gases is still incomplete.To solve that issue, Romero and Bagnato 3, 4 proposed canonical variables of work for a gas in a harmonic trap, which work like pressure and volume for a gas in a vessel and allow global measurements on the system.Therefore, we describe here our methodology to measure the thermodynamic susceptibilities of a harmonically trapped gas around the Bose-Einstein transition using the Romero-Bagnato formalism.Our experiments were divided in different phases, each having the harmonic trap with unique frequencies, related to the formalism's extensive variable.In each phase, we prepared individual samples of rubidium-87 gas in the trap and imaged them using the time-of-flight technique.Next, we fitted our images with the bimodal model and found the in situ density profiles of the samples with the standard regression procedures.Then, we determined the phase's equation of state as a function of temperature and number of atoms.For the thermodynamic analysis, we used an empirical model that we developed to fit the equation-of-state curves, allowing us to represent our data in terms of mathematical coefficients.In that way, we found the curves of internal energy, heat capacity, thermal expansion and compressibility, thus achieving a full thermodynamic description of a harmonically trapped gas.Our method and its results are unprecedented in the literature and might contribute to the further understanding of non-uniform systems, as well as the future development of quantum thermal engines.
The physical properties of a homogeneous system at the thermodynamic equilibrium are strongly constrained by its dimensionality. In this respect, the two dimensional Bose gas constitutes a particularly interesting system: while it is impossible to observe a long range order at non-zero temperatures, there exists nevertheless a phase transition to a superfluid state at low temperatures. Furthermore, owing to the reduced dimensionality, the equation of state of the weakly interacting two-dimensional Bose gas is scale invariant. In this thesis, we present an experimental study of the two-dimensional Bose gas. We measure its equation of state with two different methods, and find a good agreement with analytic and numerical predictions. These results confirm as well the scale invariance of the system. Furthermore, one of the methods allows for a determination of the equation of state with a single adjustable parameter. We then characterize the superfluid response of the system., by showing evidence for a dissipationless response of the system to a moving perturbation. Finally, we analyze the fluctuations dynamics of the 2D Bose gas, which confirms both the suppression of density fluctuations in the superfluid phase, and the dominating contribution of phonons to the phase fluctuations.
We study experimentally and numerically the equilibrium density profiles of a trapped two-dimensional 87Rb Bose gas and investigate the equation of state of the homogeneous system using the local density approximation. We find a clear discrepancy between in situ measurements and quantum Monte Carlo simulations, which we attribute to a nonlinear variation of the optical density of the atomic cloud with its spatial density. However, good agreement between experiment and theory is recovered for the density profiles measured after time of flight, taking advantage of their self-similarity in a two-dimensional expansion.
Thermodynamics of non-uniform systems has always been a challenging topic in the physical sciences. In the field of quantum gases, that subject is mainly studied with ultracold samples trapped in a harmonic potential, the first kind of confinement ever made to atomic fluids.1 Nascimbène et al.2 have already described a method to measure locally the equation of state and heat capacity of a harmonically trapped quantum gas. Nonetheless, local measurements cannot provide the volume-dependent susceptibilities of an inhomogeneous system, namely the thermal expansion and compressibility, meaning that the thermodynamic description of harmonically trapped gases is still incomplete. To solve that issue, Romero and Bagnato3, 4 proposed canonical variables of work for a gas in a harmonic trap, which work like pressure and volume for a gas in a vessel and allow global measurements on the system. Therefore, we describe here our methodology to measure the thermodynamic susceptibilities of a harmonically trapped gas around the Bose-Einstein transition using the Romero-Bagnato formalism. Our experiments were divided in different phases, each having the harmonic trap with unique frequencies, related to the formalism´s extensive variable. In each phase, we prepared individual samples of rubidium-87 gas in the trap and imaged them using the time-of-flight technique. Next, we fitted our images with the bimodal model and found the in situ density profiles of the samples with the standard regression procedures. Then, we determined the phase´s equation of state as a function of temperature and number of atoms. For the thermodynamic analysis, we used an empirical model that we developed to fit the equation-of-state curves, allowing us to represent our data in terms of mathematical coefficients. In that way, we found the curves of internal energy, heat capacity, thermal expansion and compressibility, thus achieving a full thermodynamic description of a harmonically trapped gas. Our method and its results are unprecedented in the literature and might contribute to the further understanding of non-uniform systems, as well as the future development of quantum thermal engines.
We describe a powerful method for determining the equation of state of an ultracold gas from in situ images. The method provides a measurement of the local pressure of a harmonically trapped gas and we give several applications to Bose and Fermi gases. We obtain the grand-canonical equation of state of a spin-balanced Fermi gas with resonant interactions as a function of temperature (Nascimbène et al 2010 Nature 463 1057). We compare our equation of state with an equation of state measured by the Tokyo group (Horikoshi et al 2010 Science 327 442), which reveals a significant difference in the high-temperature regime. The normal phase, at low temperature, is well described by a Landau Fermi liquid model, and we observe a clear thermodynamic signature of the superfluid transition. In a second part, we apply the same procedure to Bose gases. From a single image of a quasi-ideal Bose gas, we determine the equation of state from the classical to the condensed regime. Finally, the method is applied to a Bose gas in a three-dimensional optical lattice in the Mott insulator regime. Our equation of state directly reveals the Mott insulator behavior and is suited to investigate finite-temperature effects.
The dimensionality of a system strongly affects its physical properties; the phase transitions that take place and the type of order that arises depend on the dimension. In low dimensional systems phase coherence proves more difficult to achieve as both thermal and quantum fluctuations play a stronger role. The two-dimensional Bose fluid is of particular interest as even if full order is precluded, a residual "quasi-long" range order arises at low temperatures. Then two ingredients have a significant effect on the state of the system: (i) the finite size of a real system enables one to recover of a macroscopic occupation of a single-particle state; (ii) the interactions between particles lead to the emergence of a non-conventional type of phase transition toward a superfluid state.In this thesis, we present an experimental study of the two-dimensional (2D) Bose gas using two different energy landscapes to trap our atoms. In the first part, we use the spatial dependence of some local properties of an inhomogeneous gas to characterize the state of the equivalent homogeneous system. We extract its equation of state with a high accuracy from the gas density profiles and test its superfluid behavior by measuring the heating induced by a moving local perturbation. In the second part, we observe and characterize the emergence of an extended phase coherence in a 2D homogeneous gas in particular via a 3D-to-2D dimensional crossover. We investigate the dynamical establishment of the coherence via a rapid crossing of the dimensional crossover and observe topological defects in the final superfluid state. We compare our findings with the predictions for the Kibble–Zurek mechanism.