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Thermodynamics of a Trapped Bose Gas

Лев П. Питаевский, S. Stringari

Published Jan 1, 2016• 1 citation
Publisher

Abstract

Abstract This chapter is devoted to the thermodynamic behaviour of an interacting trapped Bose gas. It developes the Hartree–Fock theory for nonuniform interacting Bose gases at low temperatures, and discusses the problem of the shift of the critical temperature caused by interactions and also the thermodynamic behaviour of the trapped gas below the critical temperature. The chapter outlines general procedures, which do not only hold for Bose gases, to determine the equation of state of uniform matter, starting from the knowledge of the density profile in the trap. Finally, the chapter discusses some properties of the collective oscillations of a trapped Bose gas at a finite temperature.

DOI
10.1093/acprof:oso/9780198758884.003.0013

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In this chapter we consider selected topics in the theory of trapped gases at non-zero temperature when the effects of interactions are taken into account. The task is to extend the considerations of Chapters 8 and 10 to allow for the trapping potential. In Sec. 11.1 we begin by discussing energy scales, and then calculate the transition temperature and thermodynamic properties. We show that at temperatures of the order of T c the effect of interactions on thermodynamic properties of clouds in a harmonic trap is determined by the dimensionless parameter N 1/6 a /ā. Here ā, which is defined in Eq. (6.24), is the geometric mean of the oscillator lengths for the three principal axes of the trap. Generally this quantity is small, and therefore under many circumstances the effects of interactions are small. At low temperatures, thermodynamic properties may be evaluated in terms of the spectrum of elementary excitations of the cloud in its ground state, which we considered in Secs. 7.2, 7.3, and 8.2. At higher temperatures it is necessary to take into account thermal depletion of the condensate, and useful approximations for thermodynamic functions may be obtained using the Hartree–Fock theory as a starting point. The remainder of the chapter is devoted to non-equilibrium phenomena. As we have seen in Secs. 10.3–10.5, two ingredients in the description of collective modes and other non-equilibrium properties of uniform gases are the two-component nature of condensed Bose systems, and collisions between excitations. For atoms in traps a crucial new feature is the in homogeneity of the gas. This in itself would not create difficulties if collisions between excitations were sufficiently frequent that matter remained in thermodynamic equilibrium locally.

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Journal of Physics B Atomic Molecular and Optical Physics•Jun 28, 1997•22 citations
Abstract This chapter is devoted to the thermodynamic behaviour of an interacting trapped Bose gas. It developes the Hartree–Fock theory for nonuniform interacting Bose gases at low temperatures, and discusses the problem of the shift of the critical temperature caused by interactions and also the thermodynamic behaviour of the trapped gas below the critical temperature. The chapter outlines general procedures, which do not only hold for Bose gases, to determine the equation of state of uniform matter, starting from the knowledge of the density profile in the trap. Finally, the chapter discusses some properties of the collective oscillations of a trapped Bose gas at a finite temperature.
In this chapter we consider selected topics in the theory of trapped gases at non-zero temperature when the effects of interactions are taken into account. The task is to extend the considerations of Chapters 8 and 10 to allow for the trapping potential. In Sec. 11.1 we begin by discussing energy scales, and then calculate the transition temperature and thermodynamic properties. We show that at temperatures of the order of T c the effect of interactions on thermodynamic properties of clouds in a harmonic trap is determined by the dimensionless parameter N 1/6 a /ā. Here ā, which is defined in Eq. (6.24), is the geometric mean of the oscillator lengths for the three principal axes of the trap. Generally this quantity is small, and therefore under many circumstances the effects of interactions are small. At low temperatures, thermodynamic properties may be evaluated in terms of the spectrum of elementary excitations of the cloud in its ground state, which we considered in Secs. 7.2, 7.3, and 8.2. At higher temperatures it is necessary to take into account thermal depletion of the condensate, and useful approximations for thermodynamic functions may be obtained using the Hartree–Fock theory as a starting point. The remainder of the chapter is devoted to non-equilibrium phenomena. As we have seen in Secs. 10.3–10.5, two ingredients in the description of collective modes and other non-equilibrium properties of uniform gases are the two-component nature of condensed Bose systems, and collisions between excitations. For atoms in traps a crucial new feature is the in homogeneity of the gas. This in itself would not create difficulties if collisions between excitations were sufficiently frequent that matter remained in thermodynamic equilibrium locally.
The topic of Bose–Einstein condensation in a uniform, non-interacting gas of bosons is treated in most textbooks on statistical mechanics. In the present chapter we discuss the properties of a non-interacting Bose gas in a trap. We shall calculate equilibrium properties of systems in a semi-classical approximation, in which the energy spectrum is treated as a continuum. For this approach to be valid the temperature must be large compared with Δ∈/ k , where Δ∈ denotes the separation between neighbouring energy levels. As is well known, at temperatures below the Bose–Einstein condensation temperature, the lowest energy state is not properly accounted for if one simply replaces sums by integrals, and it must be included explicitly. The statistical distribution function is discussed in Sec. 2.1, as is the single-particle density of states, which is a key ingredient in the calculations of thermo dynamic properties. Calculations of the transition temperature and the fraction of particles in the condensate are described in Sec. 2.2. In Sec. 2.3 the semi-classical distribution function is introduced, and from this we determine the density profile and the velocity distribution of particles. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. The final two sections are devoted to effects not captured by the simplest version of the semi-classical approximation: corrections to the transition temperature due to a finite particle number (Sec. 2.5), and thermodynamic properties of gases in lower dimensions (Sec. 2.6).
In this chapter we consider selected topics in the theory of trapped gases at non-zero temperature when the effects of interactions are taken into account. The task is to extend the considerations of Chapters 8 and 10 to allow for the trapping potential. In Sec. 11.1 we begin by discussing energy scales, and then calculate the transition temperature and thermodynamic properties. We show that at temperatures of the order of T c the effect of interactions on thermodynamic properties of clouds in a harmonic trap is determined by the dimensionless parameter N 1/6 a/ā . Here ā, which is defined in Eq. (6.24), is the geometric mean of the oscillator lengths for the three principal axes of the trap. Generally this quantity is small, and therefore under many circumstances the effects of interactions are small. At low temperatures, thermodynamic properties may be evaluated in terms of the spectrum of elementary excitations of the cloud, which we considered in Secs. 7.2, 7.3, and 8.2. At higher temperatures it is necessary to take into account thermal depletion of the condensate, and useful approximations for thermodynamic functions may be obtained using the Hartree–Fock theory as a starting point. The remainder of the chapter is devoted to non-equilibrium phenomena. As we have seen in Secs. 10.3–10.5, two ingredients in the description of collective modes and other non-equilibrium properties of uniform gases are the two-component nature of condensed Bose systems, and collisions between excitations.
The topic of Bose–Einstein condensation in a uniform, non-interacting gas of bosons is treated in most textbooks on statistical mechanics. In the present chapter we discuss the properties of a non-interacting Bose gas in a trap. We shall calculate equilibrium properties of systems in a semi-classical approximation, in which the energy spectrum is treated as a continuum. For this approach to be valid the temperature must be large compared with Δ∊/ k , where Δ∊ denotes the separation between neighbouring energy levels. As is well known, at temperatures below the Bose–Einstein condensation temperature, the lowest-energy state is not properly accounted for if one simply replaces sums by integrals, and it must be included explicitly. The statistical distribution function is discussed in Sec. 2.1, as is the single-particle density of states, which is a key ingredient in the calculations of thermodynamic properties. Calculations of the transition temperature and the fraction of particles in the condensate are described in Sec. 2.2. In Sec. 2.3 the semi-classical distribution function is introduced. From this we obtain the density profile and the velocity distribution of particles, and use these to determine the shape of an anisotropic cloud after free expansion. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. Finally, Sec. 2.5 treats corrections to the transition temperature due to a finite particle number.
Ultracold atoms are exceptional tools to explore the physics of quantum matter. In fact, the high degree of tunability of ultracold Bose and Fermi gases makes them ideal systems for quantum simulation and for investigating macroscopic manifestations of quantum effects, such as superfluidity. In ultracold gas research, a central role is played by collective oscillations. They can be used to study different dynamical regimes, such as superfluid, collisional, or collisionless limits or to test the equation of state of the system. In this thesis, we present a unified description of collective oscillations in low dimensions covering both Bose and Fermi statistics, different trap geometries and zero as well as finite temperature, based on the formalism of hydrodynamics and sum rules. We discuss the different behaviour exhibited by the second excited breathing mode in the collisional regime at low temperature and in the collisionless limit at high temperature in a 1D trapped Bose gas with repulsive contact interaction. We show how this mode exhibits a single-valued excitation spectrum in the collisional regime and two different frequencies in the collisionless limit. Our predictions could be important for future research related to the thermalization and damping phenomena in this low-dimensional system. We show that 1D uniform Bose gases exhibit a non-monotonic temperature dependence of the chemical potential characterized by an increasing-with-temperature behaviour at low temperature. This is due to the thermal excitation of phonons and reveals an interesting analogy with the behaviour of superfluids. Finally, we investigate a gas with a finite number N of atoms in a ring geometry at T = 0. We discuss explicitly the deviations of the thermodynamic behaviour in the ring from the one in the large N limit.