We study the characteristics of Bose-Einstein condensates formed by the action of three different potentials: Newtonian self-gravity, hard-sphere repulsion (the Gross-Pittaevski potential), and a harmonic trap potential as used in laboratory experiments. We show that in a certain regime a condensate can form which is much larger than predicted by the quantum self-gravitational ground state radius. Furthermore, the size is independent of the number of particles in the condensate. We speculate that massive objects of this type may form in extreme isolation through gravitational accretion onto a fixed-size object. We propose that these be called deep-space quantum objects, or DSQOs (discos). Analytical Hartree-Fock solutions for the DSQO ground state are obtained, along with a formula for the limiting DSQO radius and energy. These are then compared with numerical calculations. Prospects for DSQO formation are briefly discussed with emphasis on possible cooling mechanisms and the dynamic stability of structures formed by accretion.
We study the characteristics of Bose-Einstein condensates formed by the action of three different potentials: Newtonian self-gravity, hard-sphere repulsion (the Gross-Pittaevski potential), and a harmonic trap potential as used in laboratory experiments. We show that in a certain regime a condensate can form which is much larger than predicted by the quantum self-gravitational ground state radius. Furthermore, the size is independent of the number of particles in the condensate. We speculate that massive objects of this type may form in extreme isolation through gravitational accretion onto a fixed-size object. We propose that these be called deep-space quantum objects, or DSQOs (discos). Analytical Hartree-Fock solutions for the DSQO ground state are obtained, along with a formula for the limiting DSQO radius and energy. These are then compared with numerical calculations. Prospects for DSQO formation are briefly discussed with emphasis on possible cooling mechanisms and the dynamic stability of structures formed by accretion.
Abstract. In the infinite three-dimensional space, we study a homogeneous nonrelativistic Schrödinger Hamiltonian describing a self-gravitating Bose gas with two types of interactions: a repulsive scaled interaction modeled by [Formula: see text] and an attractive Newtonian potential of the form [Formula: see text]. We investigate the mean-field limit as the number of particles [Formula: see text], with coupling parameters [Formula: see text] and [Formula: see text] satisfying [Formula: see text]. In this regime, we rigorously derive the corresponding Gross–Pitaevskii–Poisson theory. Furthermore, we analyze the asymptotic behavior of the condensate profile in two opposite limiting cases: when [Formula: see text], the system is well approximated by a Thomas–Fermi-type model, whereas when [Formula: see text], the condensate approaches a Choquard–Pekar-type profile.
Amplifying on a proposal by O'Dell et al. for the realization of Bose-Einstein condensates of neutral atoms with attractive 1∕r interaction, we point out that the instance of self-trapping of the condensate, without an external trap potential, is physically best understood by introducing appropriate ``atomic'' units. This reveals a remarkable scaling property: the physics of the condensate depends only on the two parameters N2a∕au and $\ensuremath{\gamma}∕{N}^{2}$, where N is the particle number, a the scattering length, au the ``Bohr'' radius, and $\ensuremath{\gamma}$ the trap frequency in atomic units. We calculate accurate numerical results for self-trapping wave functions and potentials, and for energies, sizes, and peak densities, and compare with previous variational results. We point out the existence of a second solution of the extended Gross-Pitaevskii equation for negative scattering lengths, with and without trapping potential, which is born together with the ground state in a tangent bifurcation. This indicates the existence of an unstable collectively excited state of the condensate for negative scattering lengths.
We study the ground state and the first three radially excited states of a self-gravitating Bose-Einstein condensate with respect to the influence of two external parameters, the total mass and the strength of interactions between particles. For this we use the so-called Gross-Pitaevskii-Newton system. In this context we especially determine the case of very high total masses where the ground state solutions of the Gross-Pitaevskii-Newton system can be approximated with the Thomas-Fermi limit. Furthermore, stability properties of the computed radially excited states are examined by applying arguments of the catastrophe theory.
We analytically and numerically study the ground state and collective dynamics of Bose-Einstein condensates in two traps: a Newtonian potential and a logarithmic potential inspired by Modified Newtonian Dynamics (MOND). In the ground state, the MOND potential supports bound states only in the deep-MOND regime, where the condensate becomes significantly larger than its Newtonian counterpart. The size increases with repulsive coupling parameter β in both potentials. A clear scaling law of the size with β1/3 emerges in the MOND case and is confirmed numerically over a wide parameter range, while for the Newtonian potential no simple scaling law exists as the Thomas-Fermi approximation ceases to be valid. For the dynamics, we derive and solve equations for the monopole collective mode. The larger MOND-bound condensate oscillates at a lower frequency, which scales as β−1/3 in the strong-interaction limit. These scaling laws provide insights for quantum-simulation experiments aiming to probe modified-gravity scenarios with cold atoms.
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.
Ultra-dilute self bound droplets in Bose-Einstein condensates have been of great interest since their first prediction in 2015.The self-bound nature of these bosonic gases combined with their extremely low densities make them a prime target of investigation for both experimentalists and theoreticians.In this thesis work dipole-dipole interacting gases are simulated using the Gross-Pitaevskii equation applying a fourth order, split-operator imaginary time method in two dimensions.First, the systems are studied at collapse for dipoles aligned with a magnetic field that tilts the dipoles in the two-dimensional plane.For some angles the dipole interaction becomes partly attractive which can lead to the collapse of the mean field solution.However, quantum fluctuations can stabilize the gas.Introducing the so-called Lee-Huang-Yang (LHY) correction, the system can be stabilized and self-bound droplets may form.Here, we investigate this stabilization for a quasi two-dimensional set-up. Popular science Ultra-cold droplets intrigue physicistsA hot topic within ultra-cold atom physics has been the recent discovery of droplets formed out of Bose-Einstein condensates, a specific form of a super cold and very dilute gas.The condensate is however very different from any gas encountered in everyday life.A normal gas can be thought of as atoms or molecules moving around in all directions.If you cool this gas, all the particles move slower until they either form a liquid or freeze and become a solid.This is not the case for the Bose-Einstein condensate.Instead they spread through space due to quantum mechanics and together they form one entity.This entity, the so called condensate, behaves like no classical gas, as it is mainly governed by quantum mechanics and not by the random chance of a lot of small particles moving about.To add to this strangeness, droplets now seem to form out of these condensates.The existence of these droplets was quite a surprise.Among experimentalists working with creating Bose-Einstein condensates the expected result was a collapse of the system if forces pushing the condensate together won against the forces pushing it apart.However, experimentalists revealed that some other internal force stabilized drops inside the condensate.Not only that, this force seemed to be able to create drops that could exist outside a "trap", which usually is a number of lasers and magnets that keep the condensate together.Some droplets could not be formed without a trap, but instead formed crystal structures, a very intriguing result.This has led to an excited mood within the field of cold atom physics.But what to expect from this anti-collapse force, mainly being described as quantum fluctuations, and the droplets it creates needs further study to find out.In this thesis work we simulate the anti-collapse force and possible droplet systems solving the underlying quantum-mechanical many-particle equations numerically.One system of interest is the very thin disc of condensate, or a quasi two-dimensional condensate.This system is excellent for rotating the condensate in.Why? Imagine having a piece of pizza dough that in this case represents a condensate.If you spin this dough as a skilled pizza maker does, the pizza will become flat due to rotational forces.The same goes for the condensate.As you rotate it it will become flat anyways and therefore having a model where our very cold gas is already flat makes it easier to simulate rotating systems.But before you start rotating your drops, you have to find the drops.And to find the drops you have to study the forces pushing the condensate together and the forces pushing it apart.Therefore we presently aim at studying possible droplet formation in these pizza crust shaped condensates, to make way for the study of rotational excitations.Initial results show some promise that droplets can be found in these quasi two-dimensional systems.