In this paper, we present the construction of equilibrium states for a gas of weakly interacting non-relativistic bosons, focusing on the case of a non-trivial background field in infinitely extended space. Building upon a method introduced by Araki and further developed by Fredenhagen and Lindner, we derive the generating function of the correlation functions of the theory as a suitable series. By applying a Hubbard-Stratonovich transformation, we rewrite this quantity into a more mathematically tractable form, allowing us to establish the convergence of the corresponding loop vertex expansion in certain intermediate regimes. Furthermore, we isolate the tree diagrams that produce the scattering length in the dispersion relations of the two-point function of the state within the Gross-Pitaevskii regime. Finally, we use this scattering length to renormalise the background and the two-point function of the fluctuations and we discuss convergence of the generating function of the connected correlation functions of the renormalised theory in the limit of vanishing temperature.
In this paper, we present the construction of equilibrium states for a gas of weakly interacting non-relativistic bosons, focusing on the case of a non-trivial background field in infinitely extended space. Building upon a method introduced by Araki and further developed by Fredenhagen and Lindner, we derive the generating function of the correlation functions of the theory as a suitable series. By applying a Hubbard-Stratonovich transformation, we rewrite this quantity into a more mathematically tractable form, allowing us to establish the convergence of the corresponding loop vertex expansion in certain intermediate regimes. Furthermore, we isolate the tree diagrams that produce the scattering length in the dispersion relations of the two-point function of the state within the Gross-Pitaevskii regime. Finally, we use this scattering length to renormalise the background and the two-point function of the fluctuations and we discuss convergence of the generating function of the connected correlation functions of the renormalised theory in the limit of vanishing temperature.
In this paper we present the construction of the equilibrium states at positive temperature in the presence of a condensation phase for a Gas of non relativistic Bose particles on an infinite space interacting through a localised two body interaction. We use methods of quantum field theory in the algebraic formulation to obtain this result and in order to prove convergence of the partition function and of the generating function of the correlation functions, we introduce an auxiliary stochastic Gaussian field which mediates the interaction of the Bose particles (Hubbard-Stratonovich transformation). The construction of the equilibrium state and of the partition function in the presence of the condensate, treating the auxiliary stochastic field as external potential, can be achieved using and adapting ideas and methods of Araki. Explicit formulas for the relative entropy of the equilibrium state with the external potential with respect to the equilibrium state of the free theory are obtained adapting known Feynman-Kac formulas for the propagators of the theory. If the two-body interaction is sufficiently weak, the proof of the convergence of the partition function after evaluation of the external stochastic field on a suitable Gaussian state can be given utilizing the properties of the relative entropy mentioned above. Limits where the localisation of the two-body interaction is removed are eventually discussed in combination of the limits of vanishing temperature and or in the weakly interacting regime.
Abstract This chapter develops the theory of weakly interacting nonuniform Bose gases at zero temperature. Interaction effects are accounted for by a single coupling constant fixed by the value of the s-wave scattering length. The nonlinear Gross–Pitaevskii equation is derived in both the stationary and time-dependent cases. The irrotational nature of the flow and the crucial role played by the chemical potential are highlighted. The classical nature of the equation is emphasized and the analogy with the Maxwell equations of classical electrodynamics is pointed out. The stationary solutions of the Gross–Pitaevskii equation, corresponding to quantized vortex lines, vortex rings, and solitons, are discussed. The time-dependent solutions corresponding to small-amplitude oscillations are derived and the corresponding dispersion relation is shown to coincide with the predictions of Bogoliubov theory for uniform media. The Thomas–Fermi limit is also discussed.
We study the ground state of a dilute Bose gas in a scaling limit where the Gross-Pitaevskii functional emerges. This is a repulsive non-linear Schrödinger functional whose quartic term is proportional to the scattering length of the interparticle interaction potential. We propose a new derivation of this limit problem, with a method that bypasses some of the technical difficulties that previous derivations had to face. The new method is based on a combination of Dyson's lemma, the quantum de Finetti theorem and a second moment estimate for ground states of the effective Dyson Hamiltonian. It applies equally well to the case where magnetic fields or rotation are present.
In this thesis, we apply the stochastic Gross-Pitaevskii equation, using the formulation of Stoof, to the study of weakly interacting, finite temperature atomic Bose gases, both in and out of equilibrium. Explicitly maintaining the effects of phase and density fluctuations, this state-of-the-art method is well suited to the study of low dimensional Bose systems. As such, we consider the non-equilibrium dynamics of dark matter wave solitons, oscillating within harmonically trapped, one-dimensional phase-fluctuating condensates, through numerical simulations of the stochastic Gross-Pitaevskii equation. In treating the damping due to higher lying thermal modes of the system, we include the HartreeFock contribution of the equilibrium atomic density, which is a novel feature of our work. A stochastic approach is found to be essential in capturing the shot-to-shot variation in soliton trajectories observed experimentally. Neglecting stochastic effects, we find the well-known dissipative Gross-Pitaevskii equation to recover the average dynamics of the stochastic model, on retaining the microscopically justified form for the damping of this approach. Such a coupling is found to be necessary in modelling the dissipative nature of finite temperature soliton dynamics. Furthermore, we undertake a comparison between the finite temperature equilibrium state of the stochastic Gross-Pitaevskii equation, and that predicted within a number conserving Bogoliubov approach. Through a series of numerical experiments, we compare the equilibrium properties of a harmonically trapped Bose gas, considering our results in light of those of independent theoretical treatments, and additionally assessing the sensitivity of the numerical implementation of the former method to the numerical grid spacing. In the regime in which thermal effects dominate, the stochastic Gross-Pitaevskii approach proves to be a more consistent means of generating such equilibrium states.
In this thesis interacting relativistic and non relativistic Quantum Field Theories (QFT) are studied, investigating their equilibrium configurations and their long time evolution in the framework of perturbative Algebraic Quantum Field Theory, studying also convergence in a specific model. In the first part, concerning the relativistic case, we study, in formal perturbation theory, interacting KMS states generalising results by Fredenhagen and Lindner to fermionic field theories. As an application their stability is investigated together with the characterisation of the Debye screening effect in plasmas via the explicit solution of the Semiclassical Maxwell equations. Connected to these results, we also initiate a systematic study of secular growths appearing in the truncated perturbative series and affecting the validity of perturbation theory in its long time predicting capability. We restrict to theories on Minkowski spacetime and provide, for interactions depending on time via a switching process, a sufficient condition preventing secular growths in the truncated perturbative series. In the second part, we focus on a model of non relativistic quantum field theory. We construct equilibrium states at positive temperature in the presence of a condensation phase for a gas of non relativistic Bose particles on an infinite space interacting through a localised two body interaction. The convergence is proven, both for the interacting KMS state and for the entropy of the interacting KMS state relative to the free KMS state. The strategy of our proof combines ideas of Algebraic QFT, Constructive QFT and methods of statistical mechanics in the operator algebraic setting. Limits where the localisation of the two-body interaction is removed are eventually discussed, in combination with limits of the other physical parameters of the theory such as temperature, strength of the interaction and chemical potential. Finally, in the appendix, we present results on the Araki-Uhlmann relative entropy in fermionic quantum field theories, as well as an entropy-area relation, obtained via semiclassical methods, for the cosmological horizon in the static patch of de Sitter spacetime sourced by a coherent excitation of a free quantum scalar field. This completes the collection of original results obtained during the PhD.
In the present chapter we consider the structure of the Bose–Einstein condensed state in the presence of interactions. Our discussion is based on the Gross–Pitaevskii equation, which describes the zero-temperature properties of the non-uniform Bose gas when the scattering length a is much less than the mean interparticle spacing. We shall first derive the Gross–Pitaevskii equation at zero temperature by treating the interaction between particles in a mean-field approximation (Sec. 6.1). Following that, in Sec. 6.2 we discuss the ground state of atomic clouds in a harmonic-oscillator potential. We compare results obtained by variational methods with those derived in the Thomas-Fermi approximation, in which the kinetic energy operator is neglected in the Gross–Pitaevskii equation. The Thomas–Fermi approximation fails near the surface of a cloud, and in Sec. 6.3 we calculate the surface structure using the Gross–Pitaevskii equation. Finally, in Sec. 6.4 we determine how the condensate wave function ‘heals’ when subjected to a localized disturbance. The Gross–Pitaevskii equation In the previous chapter we have shown that the effective interaction between two particles at low energies is a constant in the momentum representation, U 0 = 4πħ 2 a / m . In coordinate space this corresponds to a contact interaction U 0 δ( r – r ′), where r and r ′ are the positions of the two particles. To investigate the energy of man y-body states we adopt a Hartree or mean-field approach, and assume that the wave function is a symmetrized product of single-particle wave functions.