[Paper Review] Field theory for trapped atomic gases

H. T. C. Stoof|ArXiv.org|Oct 27, 1999

Cold Atom Physics and Bose-Einstein Condensates2 references4 citations

TL;DR

This paper presents a comprehensive quantum field theory framework for trapped ultracold atomic gases, using functional methods to describe both equilibrium and nonequilibrium phenomena. It derives microscopic foundations for Bose-Einstein condensation via Bogoliubov and Popov theories in trapped bosonic systems and for superconductivity via Bardeen-Cooper-Schrieffer theory in fermionic lithium, while extending field-theoretic techniques to nonequilibrium dynamics such as condensate growth, phase dynamics, and collective modes beyond the Gross-Pitaevskii equation.

ABSTRACT

In this course we give a selfcontained introduction to the quantum field theory for trapped atomic gases, using functional methods throughout. We consider both equilibrium and nonequilibrium phenomena. In the equilibrium case, we first derive the appropriate Hartree-Fock theory for the properties of the gas in the normal phase. We then turn our attention to the properties of the gas in the superfluid phase, and present a microscopic derivation of the Bogoliubov and Popov theories of Bose-Einstein condensation and the Bardeen-Cooper-Schrieffer theory of superconductivity. The former are applicable to trapped bosonic gases such as rubidium, lithium, sodium and hydrogen, and the latter in particular to the fermionic isotope of atomic lithium. In the nonequilibrium case, we discuss various topics for which a field-theoretical approach is especially suited, because they involve physics that is not contained in the Gross-Pitaevskii equation. Examples are quantum kinetic theory, the growth and collapse of a Bose condensate, the phase dynamics of bosonic and fermionic superfluids, and the collisionless collective modes of a Bose gas below the critical temperature.

Motivation & Objective

• To develop a unified quantum field theory framework for trapped atomic gases, applicable to both equilibrium and nonequilibrium regimes.
• To provide a microscopic derivation of the Bogoliubov and Popov theories for Bose-Einstein condensation in trapped bosonic gases like rubidium and lithium.
• To establish the Bardeen-Cooper-Schrieffer theory of superconductivity as a field-theoretic description for fermionic atomic gases, particularly lithium-6.
• To extend field-theoretic methods beyond the Gross-Pitaevskii equation to describe nonequilibrium dynamics such as condensate formation and phase coherence.
• To offer a pedagogical, self-contained introduction to functional field methods in ultracold quantum gases, suitable for advanced students and researchers.

Proposed method

• The paper employs functional integral methods throughout, using path integrals to derive effective field theories for trapped atomic systems.
• It formulates the Hartree-Fock approximation to describe the normal phase of trapped atomic gases in thermal equilibrium.
• For the superfluid phase, it derives the Bogoliubov and Popov theories from first principles using field-theoretic techniques, accounting for interactions and trapping potentials.
• It applies the same field-theoretic formalism to nonequilibrium dynamics, including quantum kinetic theory and the dynamics of phase fluctuations.
• The approach incorporates the effects of trapping potentials via a spatially dependent chemical potential and interaction terms in the action.
• It uses the Keldysh contour formalism to describe time-evolving, nonequilibrium processes such as the collapse and revival of a Bose condensate.

Experimental results

Research questions

• RQ1 How can a consistent quantum field theory be constructed for trapped ultracold atomic gases in both equilibrium and nonequilibrium states?
• RQ2 What is the microscopic field-theoretic derivation of the Bogoliubov and Popov theories for Bose-Einstein condensation in trapped bosonic systems?
• RQ3 How does the Bardeen-Cooper-Schrieffer theory emerge from a field-theoretic description in trapped fermionic atomic gases?
• RQ4 What field-theoretic tools can describe nonequilibrium phenomena such as the growth and collapse of a Bose condensate?
• RQ5 How do collective modes and phase dynamics in superfluids go beyond the scope of the Gross-Pitaevskii equation and require field-theoretic treatment?

Key findings

• The paper provides a microscopic derivation of the Bogoliubov and Popov theories for trapped Bose-Einstein condensates, showing their validity in the presence of harmonic trapping potentials.
• It establishes the Bardeen-Cooper-Schrieffer theory as a field-theoretic description of superfluidity in trapped fermionic atomic gases, particularly for lithium-6.
• The field-theoretic approach successfully describes nonequilibrium dynamics such as the time evolution of condensate fraction during collapse and revival, which are not captured by the Gross-Pitaevskii equation.
• It identifies the phase dynamics of both bosonic and fermionic superfluids as a key regime where field theory is essential, beyond mean-field and nonlinear Schrödinger approaches.
• The method reveals the existence of collisionless collective modes in a Bose gas below the critical temperature, arising from the interplay of interactions and trapping.
• The functional approach enables a systematic derivation of quantum kinetic equations and the description of relaxation processes in ultracold atomic systems.