Position- and momentum-space two-body correlations in a weakly interacting trapped condensate

We investigate the position- and momentum-space two-body correlations in a weakly interacting, harmonically trapped atomic Bose-Einstein condensed gas at low temperatures. The two-body correlations are computed within the Bogoliubov approximation and the consequences of the finite system size are highlighted in contrast to the spatially homogeneous case. In the position space, we recover the antibunching induced by the repulsive interatomic interaction in the condensed fraction localized around the trap center and the bunching in the outer thermal cloud. In the momentum space, bunching signatures appear for either equal or opposite values of the momentum and display peculiar features as a function of the momentum and the temperature. In analogy to the optical Hanbury-Brown and Twiss effect, the amplitude of the bunching signal at closeby momenta is fixed by the chaotic nature of the matter field state and its linewidth is shown to be set by the (inverse of the) finite spatial size of the associated in-trap momentum components. In contrast, the linewidth of the bunching signal at opposite momenta is only determined by the condensate size.

Figure 1

(a) Bogoliubov spectrum in a one-dimensional condensate in the Thomas-Fermi regime with $\mu /\hslash {\omega }_x=11.2\gg 1$. Numerical (black), hydrodynamic Thomas-Fermi limit (dashed red), and harmonic oscillator (dot-dashed blue) predictions for the spectrum of excitation modes as a function of the mode label $n$. [(b), (c)] Spatial profiles of the $u_n\left(x\right)$ and $v_n\left(x\right)$ functions for the $n=1$ (dashed blue line), $n=10$ (dot-dashed red line), and $n=30$ (dotted green line) modes of energies $E_{1,10,30}\simeq 1,8.59,26.0\hslash {\omega }_x$. (d) Density of the noncondensed atoms $\delta n$ for three different temperatures, $K_{B}T=0$ (dashed blue line), $K_{B}T=\mu$ (dotted red line), and $K_{B}T=2.5\mu$ (dot-dashed green line). The black solid line in panels [(b)–(d)] depicts the spatial profile $|{\mathrm{\Phi }}_0{\left(x\right)|}^2$ of the condensate [in units of ${\ell }_0^{-1}$ in panels (b) and (c) and in arbitrary units in panel (d) for readability's sake]. The plotted data have been obtained by diagonalizing the Bogoliubov operator in Eg. (4) in the text, on a grid of ${\mathcal{N}}_p=2048$ points and an integration box of size $L=80{\ell }_0$. Here, ${\ell }_0=\sqrt{\hslash /2m{\omega }_x}$ is the harmonic oscillator length.

Figure 2

Numerical solution for the two-body correlation $G_{\mathrm{\Phi }\mathrm{\Lambda }}^{\left(2\right)}(x_1,x_2)$ in real space. The panels [(a)–(d)] on the left column show color plots of this quantity for growing values of the temperature, $K_{B}T/\mu =0,1,1.5,2.5$. The panels [(e)–(h)] on the right column show cuts $G_{\mathrm{\Phi }\mathrm{\Lambda }}^{\left(2\right)}(x-\delta x,x+\delta x)$ along straight lines parallel to the antidiagonal located at different spatial positions $x=0$ (black solid line) $x=2.5{\ell }_0$ (blue dashed line), and $x=5{\ell }_0$ (red dot-dashed line) for the same values of the temperature. Numerical calculations have been performed on the same grid as in Fig. 1.

Figure 3

Numerical solution for the normalized two-body correlation $g_{\mathrm{\Lambda }\mathrm{\Lambda }}^{\left(2\right)}(x_1,x_2)$ in real space. In panel (a), we show the color plot of this quantity for the value of the temperature, $K_{B}T/\mu =2.5$. The gray central square, of side $L_{\mathrm{bec}}\approx 6.7{\ell }_0$ equal to the Thomas-Fermi radius, identifies the spatial region where the condensate is located. The panels (b)–(d) show the cuts $g_{\mathrm{\Lambda }\mathrm{\Lambda }}^{\left(2\right)}(x-\delta x,x+\delta x)$ along straight lines parallel to the antidiagonal, at the positions $x=2L_{\mathrm{bec}}$ (black circles) $x=3L_{\mathrm{bec}}$ (continuous lines), respectively, for the values of the temperature: $K_{B}T=\mu$, $K_{B}T=1.5\mu$, and $K_{B}T=2.5\mu$. We compare the numerical results with the analytical solution obtained for the classical, noninteracting gas at the same temperature (red crosses). Numerical calculations have been performed on the same grid as in Fig. 1.

Figure 4

Numerical results of the two-body correlation function $g_{\mathrm{\Lambda }\mathrm{\Lambda }}^{\left(2\right)}(k_1,k_2)$ in momentum space. Panels (a)–(c) show the profiles of $g_{\mathrm{\Lambda }\mathrm{\Lambda }}^{\left(2\right)}(k_1,k_2)$ for the values of the temperatures $K_{B}T/\mu =0,1,2.5$. The gray vertical and horizontal stripes, of width $k{\ell }_0=0.5\pi$, identify the region of the momentum space where the condensate contribution to the correlations is located. This region is not relevant for our purposes. We notice the presence of two main features in the two-body correlations in momentum space: (I) the diagonal stripe accounts for the normal contribute $g_{\mathrm{\Lambda }\mathrm{\Lambda },\mathrm{N}}^{\left(2\right)}(k_1,k_2)$ to the fourth-order correlator, and (II) the antidiagonal stripe accounts instead for the anomalous contribute $g_{\mathrm{\Lambda }\mathrm{\Lambda },\mathrm{A}}^{\left(2\right)}(k_1,k_2)$ to the fourth-order correlator. Note that the seeming increase of the linewidth of the antidiagonal stripe for large values of $|k_{1,2}|$ is a graphical artifact due to clipping of the color scale at large values of $g_{\mathrm{\Lambda }\mathrm{\Lambda }}^{\left(2\right)}$; see also Eq. (32). Such a clipping was necessary to guarantee readability of all features of the figure.

Figure 5

Numerical results of the normalized two-body correlation function $g_{\mathrm{\Lambda }\mathrm{\Lambda }}^{\left(2\right)}(k_1,k_2)$ in momentum space. Panel (a) shows how the rms of the cuts $g_{\mathrm{\Lambda }\mathrm{\Lambda },\mathrm{N}}^{\left(2\right)}(k+\delta k,k-\delta k)$ through the diagonal feature varies with the temperature, for the values of $k{\ell }_0/\pi =0.25,1,2,3,4$. At the higher temperatures (and for sufficiently high momentum values so that the Bose-Einstein statistics can be approximated by the Boltzmann one) the curves asymptotically follow the typical ${\ell }_0\sqrt{2k_{B}T/\hslash {\omega }_x}$ dependence of the nondegenerate, harmonically trapped gas (gray-dashed line). Panel (b) shows how the rms of the cuts $g_{\mathrm{\Lambda }\mathrm{\Lambda },\mathrm{N}}^{\left(2\right)}(k+\delta k,k-\delta k)$ varies with the wave vector $k$, for the values of the temperature $K_{B}T/\mu =0,0.5,1,1.5,2,2.5$. The inset shows the scaling with the temperature of the value of the plateau (red markers), compared with the analytical result in Eq. (40) relative to the noninteracting, classical gas.

Figure 6

Scaling of the rms of the cuts $g_{\mathrm{\Lambda }\mathrm{\Lambda },\mathrm{N}}^{\left(2\right)}(k+\delta k,k-\delta k)$ through the diagonal (black) and $g_{\mathrm{\Lambda }\mathrm{\Lambda },\mathrm{A}}^{\left(2\right)}(-k+\delta k,k+\delta k)$ through the antidiagonal (blue) features with the size $L_{\mathrm{bec}}$ of the condensate at $T=0$. In panels (a) and (b), we respectively show the exponent $b$ and the prefactor $a$ of the fit of the numerical data with the power law $a/L_{\mathrm{bec}}^b$.

Figure 7

Numerical results of the normalized two-body correlation function $g_{\mathrm{\Lambda }\mathrm{\Lambda }}^{\left(2\right)}(k_1,k_2)$ in momentum space. Panels (a) and (b) show the rms of the cuts $g_{\mathrm{\Lambda }\mathrm{\Lambda },\mathrm{N}}^{\left(2\right)}(-k+\delta k,k+\delta k)$ through the antidiagonal feature for the same values of the parameters as in Fig. 5.

Figure 8

Scaling of the rms of the cuts (a) $g_{\mathrm{\Lambda }\mathrm{\Lambda },N}^{\left(2\right)}(k+\delta k,k-\delta k)$ and (b) $g_{\mathrm{\Lambda }\mathrm{\Lambda },A}^{\left(2\right)}(-k+\delta k,k+\delta k)$ with the size of the condensate, for the value of the temperature $T=0$. The numerical data (symbols) are compared to the fitted curves (continuous lines), for the values of the wave vectors $k{\ell }_0/\pi =0.25,1,10$ (respectively in black dots, red triangles, and blue squares).