Josephson relation for the superfluid density and the connection to the Goldstone theorem in dilute Bose atomic gases

We derive the Josephson relation for a dilute Bose gas in the framework of an auxiliary-field (AF) resummation of the theory in terms of the normal- and anomalous-density condensates. In the leading order of the AF loop expansion (LOAF) we find two critical temperatures in the phase diagram: the critical temperature $T_c$, where the atom Bose-Einstein condensate (BEC) appears first, and the temperature $T^{☆}>T_c$, which indicates the onset of superfluidity and the appearance of a diatom condensate in the system. In this context, the Josephson relation shows that the superfluid density is related to the square of a second-order parameter, the anomalous density, $\langle A\rangle \langle \phi \phi \rangle$, which can be thought of as a diatom condensate analogous to the Cooper-pair condensate discussed in the BCS approach to dilute Fermi gases. This is in contrast with the corresponding result in the Bose gas theory without a diatom condensate, which predicts that the superfluid density is proportional to the square of the usual atom BEC condensate, $\langle \phi \rangle$. In the region between $T_c$ and $T^{☆}$ the anomalous-density Green's function contains a zero energy-momentum state corresponding to a Goldstone state. This is a two-particle state with U(1) charge two. Our findings are consistent with the prediction that in the temperature range between $T_c$ and $T^{☆}$ a fraction of the system is in the superfluid state in the absence of the usual atom BEC condensate. This situation is similar to the case of dilute Fermi gases, where the superfluid density is proportional to the square of the gap parameter. The Josephson relation relies on the existence of zero energy and momentum excitations showing the intimate relationship between superfluidity and the Goldstone theorem.

Figure 1

LOAF phase diagram. Here, ${\rho }_0={\left|\phi \right|}^2$ denotes the atom BEC condensate, $\rho$ is the density of the system, $A$ is the new order parameter related to the presence of a diatom condensate and superfluidity in the system, and $a$ is the $s$-wave scattering length.