Figure 1

Coherence length $L_{\mathrm{c}}$ scaled to the half length of the quasicondensate $L$ vs reduced temperature $T/T_{\phi }$. Comparison of experimental data [22, 39] spanning the range $T/T_{\phi }>6$ (open red triangles) to the SGPE model (upward solid black triangles). Additional numerical results generated with the SGPE model for lower temperatures $T/T_{\phi }<6$ are also shown (downward solid black triangles), together with the best fit of the SGPE results (dot-dashed black line). The horizontal experimental error bar is taken as 10$\%$ [22], based on typical experimental uncertainties, while the corresponding error in the simulated point arises from a typical $15\%$ variation in the quasicondensate atom number and 5$\%$ variation in temperature [22]; in both cases the vertical error bars fall within the symbol size. Temperatures have been scaled to $T_{\phi }\left[n_{\mathrm{qc}}\left(0\right)\right]={\hslash }^2{n}_{\mathrm{qc}}\left(0\right)/m{k}_{B}L$, where $n_{\mathrm{qc}}\left(0\right)$ is the peak quasicondensate density; in the SGPE analysis, the spatial extent of the quasicondensate is obtained ab initio as the temperature-dependent Thomas-Fermi radius $R_{\mathrm{TF}}\left(T\right)$ [88].Reuse & Permissions

Figure 2

Scaled coherence length $L_{\mathrm{c}}/L$ vs reduced temperature $T/T_{\phi }$. Comparison of experimental data Ref. [40] (open red circles) to predictions of the SGPE model (solid black squares). The reported experimental best fit (dashed red line) and the theory used in [40] to interpret the experimental data (solid red line) are also shown alongside the SGPE fit from Fig. 1 (dot-dashed black line). Typical error bars for corresponding data points in both experiment and the SGPE model are indicated, with the SGPE ones arising from a characteristic 20$\%$ variation in total atom number and an additional 5$\%$ variation in temperature (note that the SGPE error bars for the point with $T/T_{\phi }\approx 1.2$ lie within the point size and are not visible). The SGPE results are scaled to $T_{\phi }\left[n_{\mathrm{qc}}\left(0\right)\right]$ for consistency with the discussion of Fig. 1.Reuse & Permissions

Figure 3

Scaled coherence length $L_{\mathrm{c}}/L$ vs reduced temperature $T/T_{\phi }$. Comparison of numerical SGPE findings extracted via $C^{(1,\mathrm{mod})}$ (solid blue circles) and via $C^{\left(1\right)}$ (solid brown squares) against experimental data points of Ref. [40] (open red circles). The theory used in Ref. [40] is also shown (solid red line) together with the experimental best fit (dashed red line). SGPE numerical points are scaled to $T_{\phi }\left[N_{\mathrm{qc}}\right]$, and respective error bars account for $20\%$ variation in total atom number and a further 5$\%$ variation in temperature. The inset shows a comparison of SGPE results, in which temperatures are scaled to the experimentally quoted values of $T_{\phi }$. SGPE results shown in the inset are obtained by matching either the total number of atoms (blue solid circles, main figure and inset) or the quasicondensate atom number (green crosses, inset only).Reuse & Permissions

Figure 4

Scaled coherence length vs reduced temperature. Experimental data [40] (open red circles) against numerical SGPE results extracted via $C^{(1,\mathrm{mod})}$ (blue stars); in the SGPE analysis temperatures are scaled to $T_{\phi }$ extracted from the phase distribution (see the Appendix). The theory used in Ref. [40] is also shown [solid red (light gray) line] together with the reported experimental best fit [dashed red (light gray) line]. Error bars account for $20\%$ variation in total atom number [solid blue (dark gray) line] and $8\%$ variation in the standard deviation [dashed blue (dark gray) line; see the Appendix).Reuse & Permissions

Figure 5

Distribution of the phase of $a_c^{*}\psi \left(z\right)/\left|a_c\right|$ (where $a_c$ is the amplitude of the Penrose-Onsager mode) for a chosen temperature $T\approx T_{\phi }$ (left point with error bars in Figs. 2, 3, 4) at different locations from the trap center. Shown are the distributions at the condensate edge (left plot), at the point $z=L_{\phi }$ (central plot) where the distribution is fitted by a Gaussian (illustrated by the red solid curve) of standard deviation $\sigma =0.65$ [which for the particular numerical point considered occurs at $z\simeq 0.5R_{\mathrm{TF}}\left(T\right)$], and at the trap center $z=0$ (right plot). The distributions are centered and normalized, and the phase $\phi$ is scaled to $\pi$.Reuse & Permissions