Quasi-long-range order in trapped two-dimensional Bose gases

We study the fate of algebraic decay of correlations in a harmonically trapped two-dimensional degenerate Bose gas. The analysis is inspired by recent experiments on ultracold atoms where power-law correlations have been observed despite the presence of the external potential. We generalize the spin wave description of phase fluctuations to the trapped case and obtain an analytical expression for the one-body density matrix within this approximation. We show that algebraic decay of the central correlation function persists to lengths of about 20% of the Thomas-Fermi radius. We establish that the trap-averaged correlation function decays algebraically with a strictly larger exponent weakly changing with trap size and find indications that the recently observed enhanced scaling exponents receive significant contributions from the normal component of the gas. We discuss radial and angular correlations and propose a local correlation approximation which captures the correlations very well. Our analysis goes beyond the usual local density approximation and the developed summation techniques constitute a powerful tool to investigate correlations in inhomogeneous systems.

Figure 1

Selection of correlation functions for $R/\lambda =100$ and ${\eta }_0=0.15$ in terms of $\rho (r,r^{\prime },\mathrm{\Delta }\varphi )=\langle {\mathrm{\Phi }}^{*}\left(\mathbf{r}\right)\mathrm{\Phi }\left({\mathbf{r}}^{\prime }\right)\rangle$. Black data represents the sum from Eq. (6) (points) and the analytic result from Eq. (15) (solid line). The LCA from Eq. (19) is indicated by the dashed orange lines. Upper panel: Central correlation function $g_1\left(r\right)=\rho (r,0,0)$ (upper curve) decaying algebraically with exponent ${\eta }_0$, and trap-averaged correlation function ${\overline{g}}_1\left(r\right)$ (lower curve) decaying with strictly larger exponent ${\eta }_{\mathrm{trap}}>{\eta }_0$. Long-dashed red lines correspond to ${(r/\lambda )}^{-\eta }$ with $\eta ={\eta }_0$ and $\eta =1.6{\eta }_0$, respectively. The inset shows the same data on a linear scale. Lower panel: Radial correlation function $g_{\mathrm{rad}}(r,r^{\prime })=\rho (r,r^{\prime },0)$ for fixed $r/R=0.1,0.5,0.8$ (from left to right). Note that the peaks heights are $(1-r^2/R^2)$. The inset shows the angular correlation function $g_{\mathrm{ang}}(r,\mathrm{\Delta }\varphi )=\rho (r,r,\mathrm{\Delta }\varphi )$ for the same values of $r/R$ (from top to bottom). The long-dashed blue lines show $(1-r^2/R^2)\left(\right|\mathbf{r}-{\mathbf{r}}^{\prime }{|/\lambda )}^{-{\eta }_0}$, which deviates considerably.

Figure 2

Scaling exponent ${\eta }_{\mathrm{trap}}$ extracted from ${\overline{g}}_1\left(r\right)\sim {(r/\lambda )}^{-{\eta }_{\mathrm{trap}}}$ for $R/\lambda =20,100,1000$ (blue, dark red, orange). The solid lines give the result of applying Eq. (15) to the whole cloud. As a first estimate of the influence of the superfluid transition we replace the phase correlations in the outer regions of the cloud by an exponential decay with correlation length $\lambda$. This yields an increase of ${\eta }_{\mathrm{trap}}$ for small $R/\lambda$ (dashed lines). The red triangles correspond to the QMC data from Ref. [12] for the quasi-2D gas with $\tilde{g}=0.60$ and $R/\lambda \approx 50$, with ${\eta }_0$ determined from the scaling of $g_1\left(r\right)$. The error bars estimate the fitting error of ${\eta }_0$, which is substantial due to its small value.