Two-point correlations of a trapped interacting Bose gas at finite temperature

We develop a computationally tractable method for calculating correlation functions of the finite temperature trapped Bose gas that includes the effects of $s$-wave interactions. Our approach uses a classical field method to model the low energy modes and treats the high energy modes using a Hartree-Fock description. We present results of first and second order correlation functions, in position and momentum space, for an experimentally realistic system in the temperature range of $0.6T_c$ to $1.0T_c$. We also characterize the spatial coherence length of the system. Our theory should be applicable in the critical region where experiments are now able to measure first and second order correlations.

Figure 1

(Color online) Instantaneous density slice of classical field in $xz$ plane showing the system and correlation measurement geometry. The $x$ axis is indicated (white line) and two points $x$ and $x^{\prime }$ are shown. The field and density correlations between these points define $G_c^1(x,x^{\prime })$ and $G_c^2(x,x^{\prime })$, respectively. This result is for a system of $3\times {10}^5$ ${}^{87}\mathrm{Rb}$ atoms in the trap described in the text at a temperature of $159\mathrm{nK}$.

Figure 2

(Color online) Position space correlation functions of a harmonically trapped Bose gas at (a)–(c) At $93\mathrm{nK}$ and (d)–(f) at $159\mathrm{nK}$. Other parameters: (a)–(c) $N_{\mathrm{cond}}=153\times {10}^3$ (condensate number), $N_c=1.8\times {10}^5$, $N_i=1.2\times {10}^5$, ${ϵ}_{\text{cut}}=44\hslash {\omega }_z$, and $E_c=N_c\times 21.8\hslash {\omega }_z$; (d)–(f) $N_{\mathrm{cond}}=3540$, $N_c=1.8\times {10}^4$, $N_i=2.9\times {10}^5$, ${ϵ}_{\text{cut}}=36\hslash {\omega }_z$, and $E_c=N_c\times 20.5\hslash {\omega }_z$.

Figure 3

(Color online) Momentum space correlation functions of a harmonically trapped Bose gas. (a)–(c) at $93\mathrm{nK}$ and (d)–(f) at $159\mathrm{nK}$. Other parameters: (a)–(c) $N_{\mathrm{cond}}=153\times {10}^3$, $N_c=1.8\times {10}^5$, $N_i=1.2\times {10}^5$, ${ϵ}_{\text{cut}}=44\hslash {\omega }_z$, and $E_c=N_c\times 21.8\hslash {\omega }_z$; (d)–(f) $N_{\mathrm{cond}}=3540$, $N_c=1.8\times {10}^4$, $N_i=2.9\times {10}^5$, ${ϵ}_{\text{cut}}=36\hslash {\omega }_z$, and $E_c=N_c\times 20.5\hslash {\omega }_z$.

Figure 4

Coherence length $L_x$ calculated using classical field approach (circles) and semiclassical HFBP theory (line with dots). For reference ${\lambda }_{\mathrm{dB}}∕\sqrt{8\pi }$ is shown as the dashed line. System parameters as in Fig. 2.