Signature of the existence of a coherently condensed state in a dilute gas above the Bose-Einstein-condensate transition temperature

We study quantum coherence properties of a dilute gas at temperatures above, but not much above, the transition temperature of Bose-Einstein condensation. In such a gas, a small proportion of the atoms may possess coherence lengths longer than the mean neighboring-atomic distance, implying the existence of quantum coherence greater than that expected for thermal atoms. Conjecturing that a part of this proportion of the atoms may lie in a coherently condensed state, some unexplained experimental results [D. E. Miller et al., Phys. Rev. A 71, 043615 (2005)] can be explained.

Figure 1

Variation of $P_{\mathrm{con}}/\left(a_c{T}_0^{1/2}\right)$, on the logarithmic scale, with $T/T_0$ for $\gamma =2$ and $\beta =0.5$ [see Eq. (18)]. It shows an approximately exponential decay for $T$ below $3T_0$. Inset: The decay is even faster for larger $T/T_0$.

Figure 2

The $lnM$ versus the temperature $T$, for ${\tau }_p=30\mu \mathrm{s}$ and ${\lambda }_f=170\mu \mathrm{m}$. Circles: experimental results computed from Eq. (42) with $C$ replaced by the experimentally obtained contrasts $C_{\mathrm{ex}}$ given in Ref. [18]. Solid curve: analytical predictions computed from Eq. (43) with two fitting parameters, $a=0.075\mu {\mathrm{K}}^{-1.5}$ and $T_0=5.12\mu \mathrm{K}$.

Figure 3

Similar to Fig. 2, but for the contrast. The theoretical prediction (solid curve) was computed by making use of Eq. (42), with $M$ computed from Eq. (43). For comparison, $C_{\mathrm{th}}$ values of the simple thermal model are also plotted (dashed curve).

Figure 4

Similar to Fig. 2, but for ${\tau }_p=10\mu s$ with a fixed $T_0=5.12\mu K$. The fitting parameter $a=0.044\mu {\mathrm{K}}^{-1.5}$ for ${\lambda }_f=230\mu \mathrm{m}$ and $a=0.083\mu {\mathrm{K}}^{-1.5}$ for ${\lambda }_f=340\mu \mathrm{m}$.

Figure 5

Similar to Fig. 4, but for the contrast. The dashed curve represents $C_{\mathrm{th}}$.

Figure 6

Contrasts predicted by Eq. (B2), in the simple thermal model with the Bose-Einstein distribution. Dash curve: $\mu =0$. Solid curve: $\mu =-1\mu \mathrm{K}$. For comparison, predictions of Eq. (22) with the Boltzmann distribution (dotted curve) and experimental data with ${\lambda }_f=170\mu \mathrm{m},{\tau }_p=30\mu \mathrm{s}$ (filled circles) are also plotted.