Correlation evolution in dilute Bose-Einstein condensates after quantum quenches

The universal forms of quantum density and phase correlations after an interaction quench are found for dilute one-dimensional (1D), 2D, and 3D condensates. A Bogoliubov approach in a local density approximation is used. We obtain compact expressions for the most visible effects. Our results show how loss of phase coherence and antibunching are built up after the quench by quantum fluctuations. We demonstrate further that the density correlations can be observed even with imaging resolution much worse than healing length. This indicates that the direct measurement of counterpropagating atom pairs in situ in a continuum system is realistic. The conditions in contemporary 1D experiments are especially favorable for the correlation wave observations.

Figure 1

Correlations after a quantum quench from $g_0\sim 0$ to $g\gg g_0$: (a) density correlation $g^{\left(2\right)}(y,t)$ and (b) momentum distribution $n_{\mathbf{k}}\left(t\right)$ in a 1D gas, and (c) a snapshot of the phase correlation $g^{\left(1\right)}\left(\mathbf{y}\right)$ in 2D. Red (blue) indicates high (low) values. The color brown in (b) is a saturated high value, with black higher contours superposed. The healing length $\xi$, $\gamma$ parameter, distance $y$, and time unit $t_c=\hslash /m{c}^2$ are as in the text.

Figure 2

Universal solutions (8) for phase coherence scaled as $f_d^{\left(1\right)}=[g^{\left(1\right)}(y,t)-1]/\sqrt{\gamma }$, shown at long times $t=7.5t_c$.

Figure 3

Universal solutions (9) for density correlations scaled as $f_d^{\left(2\right)}=[g^{\left(2\right)}(y,t)-1]/\sqrt{\gamma }$, shown at long times $t=7.5t_c$.

Figure 4

Density correlations $g^{\left(2\right)}\left(\mathbf{y}\right)$ at $t=50t_c$ in 1D when seen with limited resolution. The number and amplitude of oscillations decrease as resolution worsens. From perfect (dark thin line), through $2\xi$, $5\xi$, to $10\xi$ (thick line).