A self-consistent mean-field (Hartree-Fock) theory for bosons for temperature T>0 is used to compute the spatial distribution of $^{7}\mathrm{Li}$ atoms in a harmonic trap. In light of recent experimental observations at Rice University, the goal is to establish the extent to which Bose-Einstein condensation (BEC) is possible with $^{7}\mathrm{Li}$ atoms, and to deduce the most effective signature of BEC from observations of the spatial profile. Our Hamiltonian, which includes the anisotropic external trap field and atom-atom interactions, is equivalent to that used previously for trapped atomic hydrogen. However, the attractive atom-atom interaction creates a localized well near the origin and precludes use of the usual local-density approximation. Eigenfunctions for the lowest orbitals are obtained by iteration. The total density is obtained from a sum with Bose population factors for these explicitly computed orbitals, together with analytic sums for unperturbed (ideal gas) orbitals for higher-lying states. For a small range of temperatures near ${\mathrm{T}}_{\mathrm{c}}$, it is found that the atomic cloud exhibits a spatially localized ``condensate peak'' composed primarily of atoms in the ground state. For lower T, such that more than 800--1200 atoms are in the ground state, the calculations become numerically unstable as found previously for T=0. Computational results for the collisional spin-flip ejection rate for T near ${\mathrm{T}}_{\mathrm{c}}$ and an estimate of the fluctuation in the number of atoms in the condensate are presented.
A self-consistent mean-field (Hartree-Fock) theory for bosons for temperature T>0 is used to compute the spatial distribution of 7Li atoms in a harmonic trap. In light of recent experimental observations at Rice University, the goal is to establish the extent to which Bose-Einstein condensation (BEC) is possible with 7Li atoms, and to deduce the most effective signature of BEC from observations of the spatial profile. Our Hamiltonian, which includes the anisotropic external trap field and atom-atom interactions, is equivalent to that used previously for trapped atomic hydrogen. However, the attractive atom-atom interaction creates a localized well near the origin and precludes use of the usual local-density approximation. Eigenfunctions for the lowest orbitals are obtained by iteration. The total density is obtained from a sum with Bose population factors for these explicitly computed orbitals, together with analytic sums for unperturbed (ideal gas) orbitals for higher-lying states. For a small range of temperatures near Tc, it is found that the atomic cloud exhibits a spatially localized ``condensate peak'' composed primarily of atoms in the ground state. For lower T, such that more than 800--1200 atoms are in the ground state, the calculations become numerically unstable as found previously for T=0. Computational results for the collisional spin-flip ejection rate for T near Tc and an estimate of the fluctuation in the number of atoms in the condensate are presented.
The mean-field properties of finite-temperature Bose-Einstein gases confined in spherically symmetric harmonic traps are surveyed numerically. The solutions of the Gross-Pitaevskii (GP) and Hartree-Fock-Bogoliubov (HFB) equations for the condensate and low-lying quasiparticle excitations are calculated self-consistently using the discrete variable representation, while the most high-lying states are obtained with a local-density approximation. Consistency of the theory for temperatures through the Bose condensation point Tc requires that the thermodynamic chemical potential differ from the eigenvalue of the GP equation; the appropriate modifications lead to results that are continuous as a function of the particle interactions. The HFB equations are made gapless either by invoking the Popov approximation or by renormalizing the particle interactions. The latter approach effectively reduces the strength of the effective scattering length asc, increases the number of condensate atoms at each temperature, and raises the value of Tc relative to the Popov approximation. The renormalization effect increases approximately with the log of the atom number, and is most pronounced at temperatures near Tc. Comparisons with the results of quantum Monte Carlo calculations and various local-density approximations are presented, and experimental consequences are discussed.
A system of charged particles in a harmonic trap is a realization of Thomson's raisin cake model. Therefore, we call it a Thomson atom. Bosonic, fermionic and mixed Thomson atoms exist. In this paper we focus on bosonic Thomson atoms in isotropic traps. Approximating the exact ground state by a condensate we investigate ground-state properties at temperature T = 0 using the Hartree-Fock theory for bosons. In order to assess the quality of our mean-field approach we compare the Hartree-Fock results for bosonic Thomson helium with an exact diagonalization. In contrast to the weakly interacting Bose gas (alkali vapours) mean-field calculations are reliable in the limit of large particle density. The Wigner regime (low particle density) is discussed.
When a gas of bosonic atoms is cooled sufficiently near to absolute zero, it undergoes a startling transition from a collection of randomly-behaved individual atoms to a phase in which thousands, even millions, of atoms act in collective unison. In this new phase, known as a Bose-Einstein condensate (BEC), the atoms all occupy the exact same quantum mechanical ground state and can be collectively described by a single phase-coherent wavefunction of macroscopic size. But how exactly does this harmonious behaviour arise within a collection of incoherent atoms as their temperature decreases? Does the condensate wavefunction grow from a single central nucleus, or does it emerge simultaneously throughout the system? And how does the situation differ when the gas is cooled suddenly, away from thermodynamic equilibrium? Can non-equilibrium behaviour be inherited by the condensate that forms? In this thesis we aim to answer these and other questions by studying the formation dynamics of Bose-Einstein condensates and the development of phase coherence. The research described herein makes use of several theoretical tools, which we lay out on the workbench in Chapter 2: Semi-classical Hartree-Fock theory provides a mean-field description of a condensate and thermal cloud at equilibrium; Quantum kinetic theory is used to calculate the dynamics of condensate growth into an equilibrium state, from a non-equilibrium thermal cloud; The stochastic projected Gross-Pitaevskii equation (SPGPE) employs a classical-field description to encapsulate the non-equilibrium dynamics of the condensate and low-energy region of the atom cloud, coupled to an equilibrium thermal cloud. This thesis is composed of four principal studies, each constituting a separate chapter.In Chapter 3 we study the formation of a Bose-Einstein condensate in a cigar-shaped 3D harmonic trap, induced by the controlled addition of an attractive ``dimple'' potential along the weak axis in a recent experiment [Garrett et al., Phys. Rev. A 83, 013630 (2011)]. In this manner condensation is induced without cooling due to a localized increase in the phase space density. We perform a quantitative analysis of the thermodynamic transformation in both the sudden and adiabatic regimes for a range of dimple widths and depths. We find good agreement with equilibrium calculations based on self-consistent semi-classical Hartree-Fock theory describing the condensate and thermal cloud. We observe there is an optimal dimple depth that results in a maximum in the condensate fraction. We also study the non-equilibrium dynamics of condensate formation in the sudden turn-on regime, finding good agreement for the observed time dependence of the condensate fraction with calculations based on quantum kinetic theory. In Chapter 4 we study the onset of phase coherence during the formation of a BEC following a sudden quench to below the transition temperature. Using the SPGPE, we simulate a recent experiment performed at ANU in which a sudden RF quench is applied to a dilute gas of metastable Helium atoms confined in an elongated 3D harmonic magnetic trap. At regularly-spaced intervals following the quench, broadband RF pulses are applied to homogeneously outcouple small (<5%) pulses of atoms, which expand and fall to a microchannel plate detector. The second-order correlation function at the detector, postulated to provide a measure of phase coherence of the in-situ cloud from which they were outcoupled, is measured versus time and compared with the condensate growth curve. A delay was observed in the relaxation of the second-order correlation function, relative to the peak in the number of condensate atoms. In our simulations we incorporate the effects of periodic outcoupling both the condensate and thermal cloud, and monitor the time-dependence of first- and second-order correlation functions in both position and momentum space. We observe long-range phase coherence in the final equilibrium state, though mild persistent phase fluctuations are present. We also observe the curious phenomenon of ``superbunching'' - an enhancement of density fluctuations - at specific values of the momentum, which we relate to anomalous correlations of fluctuations. Despite obtaining a good quantitative fit to the experimentally-measured condensate growth curve in our SPGPE simulations, we do not observe any evidence of the delay in coherence observed in the experiment, which remains unexplained. In Chapter 5 we use the SPGPE to study the crossover from a true BEC with long-range phase coherence to a quasicondensate dominated by phase fluctuations, for the case of a Bose gas in an highly-elongated 3D harmonic trap. We use parameters corresponding to an experiment in which a sudden quench resulted in the formation of a non-equilibrium condensate, the dynamics of which we study in the chapter that follows. Sweeping through a range of temperatures and chemical potentials within the crossover region, we reveal the behaviour of phase fluctuations through analysis of correlation functions. We compare the ideal-gas approximation for the transition temperature with the more accurate value obtained via the Binder cumulant. By comparing the phase coherence length with the spatial extent across which density fluctuations are suppressed, we identify the BEC/quasicondensate crossover point. We compare directly with predictions of mean-field theory [Petrov et al., Phys. Rev. Lett. 87, 050404 (2001)] and find that phase fluctuations persist to lower temperatures in our SPGPE simulations. We calculate anomalous correlations of the fluctuation field and show that the resulting correction to the mean-field condensate self-interaction energy is significant within the quasicondensate regime. In Chapter 6 we study the formation of a BEC into a non-equilibrium state by simulating an experiment in which a dilute gas of Rubidium-87 atoms in a highly-elongated 3D harmonic magnetic trap is rapidly quenched to below the transition temperature. Following the quench, a condensate is observed to form with an excessively elongated shape, and subsequently undergo damped quadrupole shape oscillations [Shvarchuck et al., Phys. Rev. Lett. 89, 270404 (2002)]. We simulate this experiment by modifying the SPGPE to incorporate a hydrodynamic thermal cloud, parametrized by axially and temporally-dependent temperature and chemical potential. Comparison against simulations without this axial dependence suggests the hydrodynamic behaviour of the thermal cloud plays a crucial role in the formation of a non-equilibrium condensate. Analyses of the first-order correlation function and the condensate wavefunction reveal the spontaneous formation of quantum vortices, as well as strong phase fluctuations at equilibrium indicating quasicondensate behaviour.
The fact that two-dimensional interacting trapped systems do not undergo Bose-Einstein Condensation (BEC) in the thermodynamic limit, though rigorously proved, is somewhat mysterious because all relevant limiting cases (zero temperature, small atom numbers, noninteracting particles) suggest otherwise. We study the possibility of condensation in finite trapped two-dimensional systems. We first consider the ideal gas, which incorporates the inhomogeneity and finite size of experimental systems and can be solved exactly. A semiclassical self-consistent approximation gives us a feel for the temperature scales; diagonalization of the one-body density matrix confirms that the condensation is into a single state. We squeeze a three-dimensional system and see how it crosses over into two dimensions. Mean-field theory, our main tool for the study of interacting systems, prescribes coupled equations for the condensate and the thermal cloud: the condensate receives a full quantum-mechanical treatment, while the noncondensate is described by different schemes of varying sophistication. We analyze the T = 0 case and its approach to the thermodynamic limit, finding a criterion for the dimensionality crossover and obtaining the coupling constant of the two-dimensional system that results from squeezing a three-dimensional trap. We next apply a semiclassical Hartree-Fock approximation to purely two-dimensional finite gases and find that they can be described either with or without a condensate; this continues to be true in the thermodynamic limit. The condensed solutions have a lower free energy at all temperatures but neglect the presence of phonons within the system and cease to exist when we allow for this possibility. The uncondensed solutions, in turn, are valid under a more rigorous scheme but have consistency problems of their own. Path-integral Monte Carlo simulations provide an essentially exact description of finite interacting gases and enable us to study highly anisotropic systems at finite temperature. We find that our two-dimensional Hartree-Fock solutions accurately mimic the surface density profile and predict the condensate fraction of these systems; the equivalent interaction parameter is smaller than that dictated by the T = 0 analysis. We conclude that, in two-dimensional isotropic finite trapped systems and in highly compressed three-dimensional gases, there is a phenomenon resembling a condensation into a single state.
This thesis reports on a study of the molecular Bose-Einstein condensation (mBEC) of the 6Li. More specifically, using model the Hartree-Fock (HF) model to fit the spatial density of the cloud, we demonstrate that the thermal cloud deplete the condensate, therefore a lower temperature is needed to condense the atoms in the ground state. Moreover the inter-atomic interactions can be controlled via Feshbach resonance (see section 2.2), allowing us to study the effects of the interactions on the thermodynamics properties of a mBEC, especially the critical temperature of the BEC that decreases when we increase the inter-atomic interactions. The description starts from the experimental procedure to produce a mBEC to its physical analysis. We draw a list describing the different chapters presented in the thesis: Chapter 1 Introduction: We present the fundamental difference between fermions and bosons, and we give some example of realisations of Bose-Einstein condensation and degenerate Fermi gas. Chapter 2 Interactions in an ultracold gas: We discuss the role of interactions depending on the type of particles, either fermions or bosons, and the phenomena of Feshbach resonance which is a powerful tool to change the interaction strength between particles. Chapter 3 Degenerate quantum gas: A brief historical section gives the fundamental ideas that initiate the long story of the Bose-Einstein condensation phenomena. We then discuss the mathematical description of a BEC without interactions, which is the starting point of the theoretical analysis used in the chapter 3 where we study the influence of the interactions. Then, we describe the statistics of Fermi gas and the different states that can be reached using Feshbach resonances, in particular the creation of the Feshbach molecules. Chapter 4 Basics of Cold atoms physics: A toolbox of the necessary knowledge to understand the underlying phenomena and concepts used in a cold atoms experiment. We describe the kind of atoms used, how these atoms interact with a magnetic and an electromagnetic field. Chapter 5 Guideline to making a 6Li BEC: This chapter describes the experimental methods used in our laboratory to cool down atoms to the regime of ultra-cold temperature T < 1µK. This section gives details about atomic beams production, the Zeemann slower (radiation force cooling), the magnetooptical trap (MOT) (cooling b molasses), optical dipole trap (evaporative cooling). Also we present the methods used to probe atomic clouds, such as in situ imaging methods. My contribution to this section has been to improve the stability of the evaporative cooling sequence (see section 5.4.4) and to optimize the shape of the the evaporative cooling ramp (see section 5.4.1) in order to maximise the phase space density (PSD), to reach Bose-Einstein condensation. Then, to make precise simulations and measurements to extract values of the trapping frequencies (see section 5.4.5) and scattering length (see section 5.5.4), the latter gives us the interaction strength of the system. These parameter are then used in the next chapter to extract thermodynamics properties of the system. Moreover, I developed the sequence for in situ absorption imaging (see section 5.5.3). Chapter 6 Study of an interacting 6Li mBEC: This chapter discusses the theory to understand the thermodynamics properties of an interacting Bose6 Einstein condensate. It is accomplished by fitting the spatial atomic density profiles with different models, the ideal gas (IG) model, the semi-Ideal (SI) gas model and the Hartree-Fock (HF) model. These three models vary in complexity from the most simple one the IG model to most elaborated one the HF model. The HF model requires a lot more of computational power than the other methods, therefore we present a new numerical method requiring less computational power proposed by Nathan Welch, and demonstrate experimentally through a statistical analysis that the HF model used to determine the thermodynamics properties of a BEC is more accurate to describe utracold atomic gas near the transition. Then, we present the results of the influence of the scattering length on the transition temperature and the chemical potential. My contribution to this section has been to develop tools to analyse the in situ atomic density profiles and improve the fitting program of Nathan (see section 6.3). Also, after taking the measurement of the in situ atomic density profiles, I developed and used the tools for the analysis of the thermodynamics properties of the mBEC (see section 6.4). I also adapted the models described in [1] to our experiment to extract the value of the radius of the condensate (see section 6.5). Finally, with N.Welch we present method based on energy conservation to verify the validity of the different model, my principal contribution to this was to modify the experiment to use this method (see section 6.7). Chapter 7 Initial work: Double well with 6Li molecules: This chapter presents the initial work to produce a double well potential to trap cold atoms and create a Josephson Junction. We present the state of the art of the different effects that have been observed, and discuss future measurements that would be achievable in our laboratory, in particular we would like to use a second specie 133Cs as impurities and observe the influence on Josephson oscillations of 6Li molecules. An experimental description of the double well trapping potential has been implemented and is also described. My contribution to this section was to design and implement the double well trapping potential.
We evaluate the zero temperature properties and Bose-Einstein condensation (BEC) of hard sphere bosons in a harmonic trap using Quantum Monte Carlo methods. The chief purpose of this work is to go beyond the dilute limit, to test the limits of the Gross-Pitaevskii equation and related mean field approximations and to explore the role of interactions in determining the zero temperature properties of the trapped Bose gas. The density is increased by adjusting both the number of trapped bosons, N , and the s-wave scattering length, a, to study systems from the highly dilute region corresponding to early experiments in trapped BEC [2, 3] up to liquid He densities and beyond. Rather than relying on the perturbative Bogoliubov approximation (which assumes a large condensate fraction) to describe the condensate, we use the one-body density matrix (OBDM) formulation of BEC so that the properties of the condensate for systems with arbitrarily large interactions may be studied. In this formulation of BEC, condensate properties are obtained by diagonalizing the OBDM and obtaining the corresponding single particle “natural orbitals” and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single particle orbital(s) with macroscopic occupation (N0 >> 1). Within this framework, we calculate the effects of interactions and increased density on the ground state energy, the density profile, the momentum distribution, the condensate fraction and condensate “wave-function” and several other properties.