Chapter 13. Bose Condensation: Reformulation

G. Benfatto, Giovanni Gallavotti

Richard A. Dunlap

This chapter discusses the way in which the cooling and trapping methods presented in the previous chapter can be used to create a Bose–Einstein condensate. This discussion begins with an overview of the basic properties of this condensate and the necessary requirements for its formation. The means by which it can be unequivocally identified are presented. Finally, some potential applications of the Bose–Einstein condensate are considered.

Pierre C. Hohenberg

Arie‐Willem de Leeuw

In this thesis we study Bose-Einstein condensation of photons. In Chapter 1 we describe the current experimental set-up and we explain the two important features of the Bose-Einstein condensate of light: dissipation through the interaction with the external reservoir of dye molecules and driving by the external pumping. In Chapter 2 we use the Schwinger-Keldysh formalism to describe the complete time evolution of the light. We show that the finite-lifetime effects of the photons due to the interaction with the dye can be characterized by a single dimensionless parameter, that depends on the power of the external laser pumping the dye. We also determine spectral functions and collective modes of the photon gas in both the normal and the Bose-Einstein condensed phases. In Chapter 3 we demonstrate that the finite-lifetime effects on the spatial first-order correlation functions of the condensed light are negligible, but are important for the temporal first-order correlations. Furthermore, we show that in the normal state the first-order correlations of the photons are suppressed by the interaction with the dye. In Chapter 4 we develop a model to describe the effect of interactions on the number fluctuations in Bose-Einstein condensates. We find good agreement between the observed number fluctuations in the light Bose-Einstein condensate and the results of our model for different interaction strenghts. In Chapter 5 we propose an experiment to observe phase diffusion in the Bose-Einstein condensate of photons and we predict the outcome of individual measurements. In our model we incorporate quantum and thermal fluctuations as well as the finite-lifetime effects. In Chapter 6 we investigate the light in the presence of a periodic potential and we study the transition from a superfluid to a Mott-insulator. We demonstrate that in this system the true Mott-insulating state is longer present due the interactions with the dye molecules. Finally, in Chapter 7 we consider Bose-Einstein condensation of light in nano-fabricated semiconductor microcavities. We demonstrate that in this system there is also a single dimensionless damping parameter that depends on the external pumping. We propose to use the experimental freedom to investigate the superfluidity of the photons via the excitation of scissors modes. We determine the frequency and damping of these scissors modes. We also calculate the density-density correlations of the excited light fluid and we show that this contains a signature of the dynamical Casimir effect.

Allan Griffin, Tetsuro Nikuni, E. Zaremba

In Chapter 3, we introduced a simple but reasonable approximation for the nonequilibrium dynamics of a Bose-condensed gas based on a generalized GP equation coupled to a kinetic equation. In Chapters 4–7, we turn to the question of how to derive such coupled equations for the condensate and noncondensate components in a way that gives a deeper understanding of the ZNG theory. Chapters 4–7 involve an introduction to Green's functions and field theoretic techniques for nonequilibrium problems. These provide the natural language and formalism to deal with the many subtle aspects of a Bose-condensed gas at finite temperatures. These four chapters are fairly technical. This chapter is mainly based on Kadanoff and Baym (1962) and Imamović-Tomasović (2001). Readers who are not interested in these questions can go straight to Chapter 8, which begins the discussion of applications of the ZNG coupled equations given in Chapter 3. Overview of Green's function approach To derive a microscopic theory of the nonequilibrium behaviour of a dilute weakly interacting Bose-condensed gas at finite temperatures, there are several different approaches available in the literature. We will use the wellknown Kadanoff–Baym (KB) nonequilibrium Green's function method. The generalization of this formalism to a Bose-condensed system was first considered by Kane and Kadanoff (1965), whose goal was to derive the Landau two-fluid hydrodynamic equations for a system with a Bose broken symmetry. The general problem consists of how to calculate the nonequilibrium response of a system induced by an external (space- and time-dependent) perturbation. In response to such an external perturbation, many interesting physical phenomena appear, including the excitation of collective modes and various transport processes.