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Scaling properties for a harmonic trapped gas near Bose-Einstein condensation using thermal global expansion coefficient measurement

E. D. Mercado-Gutiérrez, Freddy Jackson Poveda-Cuevas, Vanderlei Salvador Bagnato

Published Apr 10, 2016• 2 citations
OpenAlex

Abstract

We report the measurement of the global thermal expansion coefficient of a confined Bose gas of $^{87}\rm Rb$ in a harmonic potential around the Bose-Einstein condensation transition temperature. We use the concept of global thermodynamic variable, previously introduced and appropriated for a non homogeneous system. The data show divergence on the thermal expansion coefficient near the transition temperature and analyses of the universality behavior shows a critical exponent $\alpha = 0.15\pm 0.09$. With this obtained value and the critical exponent for the correlation length recently presented in the literature we analyse the scaling law combining dimension and critical exponents.

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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.

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Isothermal Compressibility and Thermal Expansion Coefficient for a Bose-Einstein Condensate in a Harmonic Trap

E. D. Mercado-Gutiérrez, Freddy Jackson Poveda-Cuevas, Vanderlei Salvador Bagnato

In this work we use the global thermodynamics variables to measure two thermodynamics susceptibilities such as the isothermal compressibility and the thermal expansion coefficient in a Bose-Einstein condensate of Rubidium 87.

Latin America Optics and Photonics Conference•Jan 1, 2016•0 citations
We report the measurement of the global thermal expansion coefficient of a confined Bose gas of 87Rb^{87}\rm Rb87Rb in a harmonic potential around the Bose-Einstein condensation transition temperature. We use the concept of global thermodynamic variable, previously introduced and appropriated for a non homogeneous system. The data show divergence on the thermal expansion coefficient near the transition temperature and analyses of the universality behavior shows a critical exponent α=0.15±0.09\alpha = 0.15\pm 0.09α=0.15±0.09. With this obtained value and the critical exponent for the correlation length recently presented in the literature we analyse the scaling law combining dimension and critical exponents.
Abstract The critical properties displayed by an ideal 2D Bose gas trapped in a harmonic potential are determined and characterized in an exact numerical fashion. Beyond thermodynamics, addressed in terms of the global pressure and volume which are the appropriate variables of a fluid confined in a non-uniform harmonic potential, the density-density correlation function is also calculated and the corresponding correlation length is found. Evaluation of all these quantities as Bose–Einstein condensation (BEC) is approached manifest its critical continuous phase transition character. The divergence of the correlation length as the critical temperature is reached, unveils the expected spatial scale invariance proper of a critical transition. The logarithmic singularities of this transition are traced back to the non-analytic behavior of the thermodynamic variables at vanishing chemical potential, which is the onset of BEC. The critical exponents associated with the ideal BEC transition in the 2D inhomogeneous fluid reveals its own universality class.
Ultra cold quantum gas is a convenient system to study fundamental questions of modern physics, such as phase transitions and critical phenomena. This master thesis is devoted to experimental investigation of the thermodynamics susceptibilities, such as the isothermal compressibility and the thermal expansion coefficient of a trapped Bose-Einstein condensate (BEC) of 87Rb atoms. The critical phenomena and the critical exponents across the transition can explain the behavior of the isothermal compressibility and the thermal expansion coefficient near the critical temperature TC. By employing the developed formalism of global thermodynamics variables, we carry out a statistical treatment of Bose gas in a 3D harmonic potential. After that, comparison of obtained results reveals the most appropriate state variables describing the system, namely volume and pressure parameter V and Π respectively. The both are related with the confining frequencies and BEC density distribution. We apply this approach to define the set of new thermodynamic variables of BEC, and also to construct the isobaric phase diagram V T. Its allows us to extract the compressibility κT and the thermal expansion coefficient βΠ. The behavior of the isothermal compressibility corresponds to the second-order phase transition, while the thermal expansion coefficient around the critical point behaves as β ∼ tr-α, where tr is reduced temperature of the system and α is the critical exponent on the basic of these. Results we have obtained the critical exponent α = 0.15±0.09, which allows us to determine the system dimensionality by means of the scaling theory, relating the critical exponents with the dimensionality. As a result, we found out that the dimensionality of the system to be d ∼ 3, one is in agreement with the real dimension of the system.
The phase transition of Bose-Einstein condensation was studied in the critical regime, where fluctuations extend far beyond the length scale of thermal de Broglie waves. We used matter-wave interference to measure the correlation length of these critical fluctuations as a function of temperature. Observations of the diverging behavior of the correlation length above the critical temperature enabled us to determine the critical exponent of the correlation length for a trapped, weakly interacting Bose gas to be nu = 0.67 +/- 0.13. This measurement has direct implications for the understanding of second-order phase transitions.
We consider a Bose gas confined by a harmonic potential. We define an appropriate thermodynamic limit and analyze the properties of the phases and phase transition in this limit. Critical properties in the presence of the potential are found to be different from, though simply related to, those in the usual translationally invariant case. We argue that the properties of magnetically trapped rubidium and sodium gases (in which Bose-Einstein condensation has recently been observed) are well approximated by our thermodynamic limit except in a narrow window of temperature around the critical temperature. We also consider the effect of the confining potential on the non-equilibrium dynamics following a rapid quench to the ordered side and give a scaling description of the late-time universal dynamics.
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.