Quantum-statistical mechanics of an atom-dimer mixture: Lee-Yang cluster expansion approach

We use the Lee-Yang cluster expansion method to study quantum-statistical properties of a mixture of interconvertible atoms and dimers, where the dimers form in a two-body bound state of the atoms. We point out an infinite series of cluster diagrams whose summation leads to the Bose-Einstein condensation of the dimers below a critical temperature. Our theory captures some important features of a cold atom-dimer mixture such as interconversion of atoms and dimers and properties of the mixture at the unitarity limit.

Figure 1

Schematic diagrams of cluster expansions. (a) One-particle connected diagram, where each dot represents $U_1$ which describes the quantum exchange effect. (b) An example of the connected diagrams of two-particle clusters, where each dot represents $U_2$ which describes the exact two-body scattering but not the quantum exchange effect. We distinguish $U_1$ and $U_2$ by the number of the incoming and outgoing lines. (c) A thick line represents the sum of all one-particle connected diagrams. A mathematical expression of this replacement is given in Eq. (27). (d) Corrections to the two-body clusters due to the quantum exchange effect. Such corrections can be taken into account through the replacement of thin lines by thick ones.

Figure 2

Dependence of $p∕\rho k_{\mathrm{B}}T$ on $x={\lambda }_{\mathrm{dB}}∕\sqrt{2\pi }a$ for $\rho {\lambda }_{\mathrm{dB}}^3=0.001,0.01,0.1$. With increasing $x$, the pressure decreases to about one half of that of the ideal atomic gas because of dimer formation.