Historical research guide · 0517 · 2013
Bose–Einstein condensation versus Dicke–Hepp–Lieb transition in an optical cavity
Francesco Piazza · Philipp Strack · Wilhelm Zwerger
In one minute
Cool a gas of atoms inside a mirrored cavity, shine a laser across it, and two quite different things can happen. The gas can become a Bose–Einstein condensate, every atom sharing one quantum state. Or it can self-organise: the atoms arrange themselves into a checkerboard whose spacing matches the cavity’s own light wave, because that arrangement scatters pump light into the cavity best. The second one is the Dicke–Hepp–Lieb transition. Francesco Piazza, Philipp Strack and Wilhelm Zwerger solve the two together exactly for an ideal gas at any temperature and map the whole phase diagram, including a special point where the gas condenses and orders itself at the same moment. Two results stand out. The lattice that the cavity light builds pushes the condensation temperature down. And warming the gas can make it order more readily rather than less — which the simplified Dicke picture gets backwards. The optical potential doing all of this, the authors write, is generated by the electromagnetic vacuum field of the resonator.
Connection
Chapter 2 argues that a cavity reshapes the vacuum’s own field and that matter answers; this paper is the exact many-body accounting of that answer, in a system where light and atoms build each other’s structure. Chapter 11 cares about coherent quantum matter under a field, and the authors place this problem in the same exactly solvable family as the BCS theory of superconductivity.
From the source
The paper gives an exact solution for the interplay between Bose-Einstein condensation and the Dicke-Hepp-Lieb self-organisation transition of an ideal Bose gas trapped in a single-mode optical cavity and driven by a transverse laser, using an effective action approach with no semi-classical approximation and no truncation of the atomic Hilbert space.Abstract; Section I A, Key results; Section III, Effective action approach
The full phase diagram at arbitrary temperature contains four phases — thermal or condensed, each either homogeneous or spatially ordered at the cavity wave vector — and a bi-critical point where the transitions cross and the atoms become superfluid and self-organise at the same time.Section I A, result i; Section IV A and Figure 3; Conclusions, Section VII
The usual truncation of the atoms to two momentum states, which reduces the problem to an effective Dicke model, is correct only for an ideal Bose gas at zero temperature: at any finite temperature the continuum of atomic momenta, and at zero temperature any appreciable repulsive interaction, makes that truncation unphysical.Section I, Introduction, the paragraph on the two-mode approximation; Section II B; Conclusions, Section VII
The cavity’s backaction generates a dynamical, temperature-dependent band structure for the atoms, and as that lattice deepens the critical temperature for Bose-Einstein condensation is strongly suppressed.Section I A, result ii; Section IV C, Dynamical band structure
The self-organisation threshold is non-monotonic in temperature: raising the temperature from zero lowers the threshold over a range, so thermal fluctuations can enhance the tendency of the atoms to arrange themselves periodically — in striking contrast with the finite-temperature Dicke model, where temperature always counteracts the ordering. The effect comes from the thermal occupation of the momentum continuum, entering through the bosonic Lindhard function and the condensate occupation, and the Dicke prediction is recovered only in the very low and very high temperature limits.Abstract, final sentence; Section IV B, Self-organization threshold, and Figures 3 and 4; threshold expression, Equation 29
Structurally, the cavity-Bose gas problem without short-range interactions between the atoms belongs to the class of exactly solvable restricted mean-field models, the most prominent of which is the BCS theory of superconductivity, and the mean-field solution becomes exact in the limit of large atom number; the authors name the open extensions — lossy cavities, the separation of temperature-induced from decay-induced dissipation, and interacting Bose gases in the superfluid regime.Section VI, Mean-field nature of the problem; Conclusions, Section VII, the structural-insights paragraph and the closing paragraph
The way in
https://doi.org/10.1016/j.aop.2013.08.015How to cite it
Francesco Piazza, Philipp Strack, Wilhelm Zwerger (2013) Bose–Einstein condensation versus Dicke–Hepp–Lieb transition in an optical cavity doi:10.1016/j.aop.2013.08.015