Second-order virial expansion for an atomic gas in a harmonic waveguide

The virial expansion for cold two-component Fermi and Bose atomic gases is considered in the presence of a waveguide and in the vicinity of a Feshbach resonance. The interaction between atoms and the coupling with the Feshbach molecules is modeled using a quantitative separable two-channel model. The scattering phase shift in an atomic waveguide is defined. This permits us to extend the Beth-Uhlenbeck formula for the second-order virial coefficient to this inhomogeneous case.

Figure 1

Example of the contour of integration in Eq. (71), where $E_1$ and $E_2$ are the energies of two-bound states.

Figure 2

(a) Vertex associated with the direct interaction in the finite-temperature formalism: $g\langle {\alpha }_1{\alpha }_2\left|{\widehat{A}}_{\epsilon }^{†}{\widehat{A}}_{\epsilon }\right|{\alpha }_3{\alpha }_4\rangle \delta \left(\tau \right)$. (b) Idem for the interchannel coupling: $\mathrm{\Lambda }\langle {\alpha }_m\left|{\widehat{A}}_{\epsilon }\right|{\alpha }_1{\alpha }_2\rangle \delta \left(\tau \right)$.

Figure 3

Diagrammatic form of the Bethe-Salpeter equation for the atom-atom vertex ${\widehat{\mathrm{\Gamma }}}^{\mathrm{a}}$.

Figure 4

Diagrammatic form of Eq. (95).

Figure 5

The two diagrams contributing to $\mathrm{\Delta }{b}_{1,1}$; see Eqs. (97) and (98).