One can derive a formula for the entropy and the free energy density in terms of the full distribution of particles over all the energy levels in the thermodynamic limit. The singular part of the measure m does not contribute to the entropy in the thermodynamic limit. The HYL model is obviously still unsatisfactory for explaining the properties of liquid. However, recently Bose-Einstein condensation has been observed in other, more artificial systems. These are produced by confining a very rare gas by means of laser beams. This technique was invented by S. Chu, C. Cohen-Tannoudji, and W. D. Phillips, who earned the 1997 Nobel prize for their contributions. The density in these systems is much lower than that of liquid 4He and as a result the interaction between the atoms is much smaller. The first experimental evidence for Bose-Einstein condensation in this kind of system was obtained in 1995 by M. H. Anderson et al.
One can derive a formula for the entropy and the free energy density in terms of the full distribution of particles over all the energy levels in the thermodynamic limit. The singular part of the measure m does not contribute to the entropy in the thermodynamic limit. The HYL model is obviously still unsatisfactory for explaining the properties of liquid. However, recently Bose-Einstein condensation has been observed in other, more artificial systems. These are produced by confining a very rare gas by means of laser beams. This technique was invented by S. Chu, C. Cohen-Tannoudji, and W. D. Phillips, who earned the 1997 Nobel prize for their contributions. The density in these systems is much lower than that of liquid 4He and as a result the interaction between the atoms is much smaller. The first experimental evidence for Bose-Einstein condensation in this kind of system was obtained in 1995 by M. H. Anderson et al.
Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum effects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an effective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in different, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in L2-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for sufficiently low densities as predicted in the physics literature first by Wu and subsequently others. Full-text embargoed until: 2026-01-01
Liquid helium is the prototypical example of a superfluid – a liquid that flows without viscosity and transfers heat without a temperature gradient. These properties are intimately related to the Bose condensation that occurs in this strongly interacting liquid. Bose condensation is most directly observed in the single particle atomic momentum distribution, where the Bose condensate appears as a delta function singularity. In this article, we discuss the experimental techniques used to observe the condensate and the current status of measurments of the Bose condensate in liquid helium. Introduction Liquid helium ( 4 He) has fascinated physicists ever since Kammerlingh-Onnes liquified the last of the so called permanent gases in 1908. However, evidence of what is without doubt the most fascinating property of this unique liquid, superfluidity, was not reported until almost 25 years later [1]. The superfluid phase, where heat is transferred without a thermal gradient and mass flows without a driving pressure, is a macroscopic manifestation of microscopic quantum effects governing the behavior of atoms [2]. Bose–Einstein condensation was first proposed by London [3] as the microscopic explanation for these fascinating phenomena. Helium is unique among condensed atomic systems since the bulk properties of the liquid are dominated by quantum effects. The most important of these are the statistical effects that arise for identical particles. For bosons, such as 4 He atoms, there is no limitation on the number of particles that can occupy a single quantum state.
In 1995 two different research groups observed for the first time the Bose-Einstein condensation (BEC) in ultracold gases. When the confining magnetic trap was turned off the gas was left free to expand, and the velocity of the particles showed a clear peak: most of the particles were occupying the same single particle state, the one of lowest energy. The Bose-Einstein condensation had been predicted in 1925 by Einstein, written by inspiration of a work on the statistic of the photons by Bose (1924). In this work Bose described the behavior of an ensemble of photons, treating them as massless particles, with no number conservation associated. Einstein extended this approach to particles with a mass and with fixed number, creating what is now called the Bose-Einstein distribution. The particles that follow such a description are called ``bosons'', as opposed to the ``fermions'' of the Fermi-Dirac statistics. Einstein predicted that in a gas of bosons - under a critical temperature - a finite fraction of the total number of particles would have been in the ground state, and act as a single entity. This amusing theoretical discovery found its utility a few years later. In the late thirties, new techniques allowed to cool Helium-4 at few Kelvins above the absolute zero. The properties of the resulting liquid were a puzzlement to the scientific community: among others, it could flow without experiencing friction. The liquid was called a ``superfluid''. A first explanation was given by London in 1938, which linked the superfluid behavior to the presence of a BEC among the bosonic Helium particles. The fermions cannot condense by themselves. On the other hand, they can form bound pairs and act as bosons, as it happens in a metal at low temperature. Using this approach, in 1957 Bardeen, Cooper and Schrieffer created a successful model of superconductivity by describing a superconductor as a superfluid in a charged system. During the course of these years we explored the superfluid properties of Bosons and Fermions in different settings. The original contributions of the thesis are described starting from the third chapter, where we speak about the generation and stability of solitons in a periodic optical lattices, both fixed or in motion. In the fourth chapter we study the generation of giant vortices in cold fermions, by using a generalized hydrodynamical approach. In chapter 5 we study the effect of a quasiperiodic lattice and the glassy phase it produces on a gas of bosons. Finally, we study the interaction of normal matter and superfluids with gravitational waves. While this interaction is seen to be extremely small, we believe that the resulting formalism is interesting by itself.
Just a couple decades ago bosonic low-temperature physics was synonymous with one particular liquid at a temperature of several degrees Kelvin. This liquid, helium-4, and its famous Helium-II phase, has revealed remarkable quantum properties such as quantized vortices and second sound. Yet, despite many things learned by investigating helium, its study has certain limitations. On the one hand, helium is a strongly interacting liquid, which makes first-principles theoretical research challenging. To name one example, establishing a connection between the non-interacting condensate state described by Bose and Einstein, and the helium properties observed in the lab took decades. Furthermore, helium is a specific chemical element, and hence there is no surprise that its physical properties are fixed by nature. However, every particular chemical element has particular physical properties, so for a long time it seemed that it was as good as it gets. The picture has changed drastically when the first Bose-Einstein condensate (BEC) was observed in an ultracold atomic gas. “Low temperature” has become synonymous with “ultracold”, and the latter implies temperatures on the order of nanokelvins. Most strikingly, the relationship between what is fixed by nature and what is experimentally changeable has evolved. In particular, systems of different particle statistics, spin, and interactions have been engineered using ultracold vapours of various alkali atoms. This versatile manipulation not only makes emulating various condensed-matter physics models in a controlled system without impurities possible, but also opens a path towards novel phenomena that are at present not achievable in any other manner. In this thesis, we have explored some of the novel phenomena that arise in systems of bosonic particles with internal degrees of freedom, such as (pseudo)spin and electric dipole moment. In the introduction, we describe a ferromagnetically coupled spin-1/2 Bose gas with contact interactions in the mean- field approximation. In particular, we compute and discuss the phase diagram of this gas, since it is a simple system which nevertheless has two order parameters. In the chapters that follow, the ideas touched upon in the introduction are developed in more detail. In Chapter 2 we develop a hydrodynamic description of the ferromagnetic spin-1/2 Bose gas at arbitrary temperatures. We study magnetization relaxation and geometric forces. In particular, we consider the topological Hall effect due to the presence of a skyrmion. In Chapter 3 we investigate the miscible (non-ferromagnetic) spin-1/2 Bose gas at arbitrary temperatures, construct its hydrodynamic description, calculate the thermodynamic properties, and study the collective modes of this system. In Chapter 4 we discuss the influence of the off-diagonal Berry curvature on the Bose-Einstein condensation temperature. Finally, in Chapter 5 we study a Bose-Einstein condensate of dipolar molecules in a weak electric field and find it to be described by a quantum rotor model. Moreover, we show that the molecular Bose-Einstein condensate is a ferroelectric material that is fully disordered by quantum fluctuations.
The refinement and the combination of laser cooling and evaporative cooling in dilute systems of alkali atoms lead to the achievement of Bose-Einstein condensation in 1995. Nowadays there is a fruitful interplay between theoretical and experimental results: in this context, the study of uniform Bose gases plays an important role in the comprehension of the static and dynamic properties of a bosonic system. Here we use quantum field theory, within the approach of functional integration, to study an homogeneous system of weakly-interacting bosonic cold atoms. After characterizing their thermodynamical properties, we derive - at a gaussian level - the number density and the superfluid density in three and in two spatial dimensions, expressed as functions of the condensate density and the temperature. Since the equations derived are valid for a generic interaction potential, we implement these formulas for bosons with a zero-range interaction, providing also an extension of these results with the inclusion of a finite-range interaction.
The Bose-Einstein condensation mechanism is shown to be capable of accounting for the existence of separate hydrodynamic velocity fields for the normal and the superfluid, provided suitable assumptions are made with regard to the single-particle energy spectrum. We consider the effect on the microcanonical gas distribution of imposing a nonzero value of the total momentum P. For an ordinary gas the effect is trivial, the whole distribution being merely shifted in momentum space. However, in the case of a degenerate Bose gas whose single-particle energy spectrum has a sharp minimum (gap or cusp), only the excited part of the gas (normal fluid) participates in the imposed motion, the condensate (superfluid) remaining "frozen" in momentum space. This rigidity of the condensate in momentum space plays the same role as the rigidity of the superelectrons on imposition of a magnetic field in the London theory of superconductivity. P acts as an additional thermodynamic variable, states with P \ensuremath{\ne} 0 being macroscopically metastable and corresponding to the existence of a relative velocity between normal and superfluid. The basic hydrodynamic assumption of the two-fluid model is thus reduced to an assumption concerning the form of an effective single-particle energy spectrum, and the parallelism between the theories of superconductivity and superfluidity is clearly exhibited. The present theory permits, in particular, the introduction of the mathematical form of Landau's phonon and "roton" spectra within the framework of the Bose-Einstein condensation picture. The statistical-thermodynamic formulas are derived and are shown to lead to characteristic two-fluid equations derived previously from a variational principle.